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# Sum of permutations!

I just found that sum of $$n$$ permutations with $$i$$ equals $$e\Gamma(n+1, 1)$$. In other words,

$$P(n,0) + P(n,1) + P(n, 2) + .... P(n,n) = e\Gamma(n+1, 1)$$. [where $$\Gamma(x,y)$$ is the incomplete gamma function].

Can anyone give a proof of this?

Note by Kartik Sharma
1 year, 11 months ago

Sort by:

Hi Kartik Sharma , see this or the solution to this question .

But if you aren't familiar with Gamma function , see the link I provided or see it here .

Hope I was useful !!! · 1 year, 11 months ago

Thanks, that was helpful. You are quite good, solves almost all the problems. · 1 year, 11 months ago

You are welcome. · 1 year, 11 months ago


text:
# Special PDE/Analysis Seminar

Wednesday, October 31, 2018 at 4:00pm

2-131

Vlad Vicol (Courant)

Solving the Navier-Stokes equation on the complement of a thin set

We construct wild solutions to the Navier-Stokes equations which have bounded kinetic energy, integrable vorticity, and which are smooth almost everywhere. In fact, these solutions are smooth outside a fractal set of times with Hausdorff dimension strictly less than one. This is based on joint work with T. Buckmaster and M. Colombo.

Note different weekday and room!

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The Interplay Between Stability and Regret in Online Learning

This paper considers the stability of online learning algorithms and its implications for learnability (bounded regret). We introduce a novel quantity called forward regret that intuitively measures how good an online learning algorithm is if it is allowed a one-step look-ahead into the future. We show that given stability, bounded forward regret is equivalent to bounded regret. We also show that the existence of an algorithm with bounded regret implies the existence of a stable algorithm with bounded regret and bounded forward regret. The equivalence results apply to general, possibly non-convex problems. To the best of our knowledge, our analysis provides the first general connection between stability and regret in the online setting that is not restricted to a particular class of algorithms. Our stability-regret connection provides a simple recipe for analyzing regret incurred by any online learning algorithm. Using our framework, we analyze several existing online learning algorithms as well as the "approximate" versions of algorithms like RDA that solve an optimization problem at each iteration. Our proofs are simpler than existing analysis for the respective algorithms, show a clear trade-off between stability and forward regret, and provide tighter regret bounds in some cases. Furthermore, using our recipe, we analyze "approximate" versions of several algorithms such as follow-the-regularized-leader (FTRL) that requires solving an optimization problem at each step.

Authors

• 6 publications
• 51 publications
• 47 publications
• Competing With Strategies

We study the problem of online learning with a notion of regret defined ...
02/12/2013 ∙ by Wei Han, et al. ∙ 0

• Stability Conditions for Online Learnability

Stability is a general notion that quantifies the sensitivity of a learn...
08/16/2011 ∙ by Stéphane Ross, et al. ∙ 0

• Online Learning with Imperfect Hints

We consider a variant of the classical online linear optimization proble...

• A Modular Analysis of Adaptive (Non-)Convex Optimization: Optimism, Composite Objectives, and Variational Bounds

Recently, much work has been done on extending the scope of online learn...
09/08/2017 ∙ by Pooria Joulani, et al. ∙ 0

• Fully Implicit Online Learning

Regularized online learning is widely used in machine learning. In this ...
09/25/2018 ∙ by Chaobing Song, et al. ∙ 8

• Online convex optimization and no-regret learning: Algorithms, guarantees and applications

Spurred by the enthusiasm surrounding the "Big Data" paradigm, the mathe...
04/12/2018 ∙ by E. Veronica Belmega, et al. ∙ 0

• Online Learning with Continuous Variations: Dynamic Regret and Reductions

We study the dynamic regret of a new class of online learning problems, ...
02/19/2019 ∙ by Ching-An Cheng, et al. ∙ 8

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1 Introduction

The fundamental role of stability in determining the generalization ability of learning algorithms in the setting of iid data is now well recognized. Moreover, our knowledge of the connection between stability and generalization is beginning to achieve a fair degree of maturity (see, for instance, [4, 13, 18, 22]). However, the same cannot be said regarding our understanding of the role of stability in online adversarial learning.

Recently, several results have shown connections between learnability of a concept class and stability of its empirical risk minimizer (ERM). Apart from theoretical interest, such insights into stability and learnability, can potentially help in designing more practical algorithms. For example, [13] show that under certain settings, stability is a more general characterization than VC-dimension; good generalization performance can be guaranteed for concept classes with stable ERM, even if its VC-dimension is infinite.

However, most of the existing implications of stability are in the batch or i.i.d. learning setting, with only a few results in the online adversarial setting. Online learning can be modeled as a sequential two-player game between a player (learner) and an adversary where, at each step, the player takes an action from a set and the adversary plays a loss function. The player’s loss is evaluated by applying the adversary’s move to the player’s action and key quantity to control is the

regret of the player in hindsight. Understanding stability in the online learning setting is not only a challenging theoretical problem but is also important from the point of view of applications. For instance, stability allows us to derive guarantees that apply to dependent (non-iid) data [2] and is critical in areas such as privacy [11].

There is a fundamental challenge in extending the connection between stability and learnability from the iid to the online case. In the iid setting, empirical risk minimization (ERM) serves as a canonical learning algorithm [23]. Thus, given any hypothesis class, it is sufficient to just analyze the stability of ERM over the class to characterize its learnability in the batch setting. Unfortunately, no such canonical scheme is known for online learning, making it significantly more involved to forge connections between online learnability and stability. We circumvent this difficulty by studying connections between stability and regret of arbitrary online learning algorithms.

In this paper, we circumvent the above mentioned issue by studying connections between stability and regret of learning algorithms, rather than online learnability of individual concept classes. in a generic sense.To this end, we first define stability for online learning algorithms. Our definition is essentially “leave last one out” stability, also considered by [20]. We also define a uniform version of this stability measure. However, stability alone cannot guarantee bounded regret. For example, an algorithm that always plays one fixed move is clearly the most stable any algorithm can be. But its regret can hardly be bounded. Hence, an additional condition is required that forces the algorithm to make progress. To this end, we introduce a novel measure called forward regret: the excess loss incurred with a look-ahead of one time step (i.e., when player makes its move after seeing the adversary’s move). We show fundamental results relating the three conditions, namely online stability, bounded forward regret and bounded regret. First, assuming stability, bounded regret and bounded forward regret are equivalent. Second, given an algorithm with bounded regret, we can always obtain a stable algorithm with bounded regret and bounded forward regret. We would like to stress that these general results do not rely on convexity assumptions and are not restricted to a particular family of learning algorithms. In contrast, [20] provides equivalence of stability and regret for only certain families of algorithms and concept classes.

We illustrate the usefulness of our general framework by considering several popular online learning algorithms like Follow-The-Leader (FTL) [10, 7], Follow-The-Regularized-Leader (FTRL) [17, 1], Implicit Online Learning (IOL) [12], Regularized Dual Averaging (RDA) [24] and Composite Objective Mirror Descent (COMiD) [8]. We obtain regret bounds for all of them using the fundamental connections between forward regret and stability thereby demonstrating that our framework is not restricted to a particular class of algorithms. Our regret analysis is arguably simpler than existing ones and, in some cases such as IOL, provides tighter guarantees as well.

Finally, we consider “approximate” versions of RDA, IOL, and FTRL algorithms where the optimization problem at each step is solved only up to a small but non-zero additive error. It is important to consider such an analysis because, in practice, the optimization problems arises at each step will not be solved to infinite precision. For each of these three algorithms, we use our general stability based recipe to provide regret bounds for their approximate versions.

We introduce our setup in Section 2. We introduce the online learning framework in Section 3 and review existing work and contrast it to our work in section 4. We introduce our three online learning conditions and show their connections in Section 5. We provide several illustrations of the usefulness of our conditions in analyzing existing online algorithms in Section 6 and finally conclude with Section 9.

2 Bregman Divergences and Strong Convexity

Here we recall the definition of a Bregman divergence [5, 6] which finds use in online learning algorithms. We also relate it to the notion of strong convexity, a key property behind many regret bounds for online learning.

Definition 1.

Let be a strictly convex function on a convex set . Also, let be differentiable on the relative interior of , , assumed to be nonempty. The Bregman divergence generated by the function is given by

 DR(x,y)=R(x)−R(y)−∇R(y)⊤(x−y)

where is the gradient of the function at .

Definition 2.

A convex function is strongly convex with respect to a norm if there exists a constant such that

 Df(u,v)≥α2∥u−v∥2∀u,v∈Rd.

is called the modulus of strong convexity and is also referred to as -strongly convex.

Now, we present a useful lemma characterizing optima of a strongly convex function.

Lemma 3.

Let be an -strongly convex function and let be a convex set. Let be a minimizer of over , i.e., . Then, for any ,

 f(u)≥f(w∗)+α2∥u−w∗∥2.

In particular, the minimizer is unique.

Lower bold case letters (e.g., ,

) denote vectors,

denotes the -th component of . The Euclidean dot product between and is denoted by or . A general norm is denoted by and refers to its dual norm. For most of this paper, we work with arbitrary norms and we use to refer to a specific norm. Unless specified otherwise, , is a compact convex set, and is any loss function. A function is -Lipschitz continuous w.r.t. a norm if .

3 Setup

We now describe the online learning setup that we use in this paper. Let be a fixed set and be a class of real-valued functions over . Now, consider a repeated game of rounds played between a player/learner and an adversary. At every step ,

• The player plays a point from a set .

• The adversary responds with a function .

• The player suffers loss .

The quantity of interest in online learning is the regret which measures how good the player performs compared to the best fixed move in hindsight (i.e. knowing all the moves of the adversary in advance). Regret is defined below in (6). The goal in online learning is to minimize the regret regardless of the function sequence played by the adversary. Online Convex Programming (OCP) [25]

(respectively Online Linear Programming (OLP)) is a special case of the online learning game above where the set

is a compact convex set and is a class of convex (respectively linear) functions defined on .

4 Related Work

For a general introduction to online learning and descriptions of standard algorithms, see [7]. In the iid setting, stability is investigated from various points of view in [4, 13, 18, 22]. There are only a few papers dealing with stability in the online setting. Recently, [20] defined what we call Last Leave-One-Out (LLOO) stability and showed that for FTRL or MD type methods, stable online learning algorithms have bounded regret. In contrast, we distill out the “progress” in terms of forward regret condition and show a much more general connection between stability, regret and forward regret. Unlike [20], our method is extremely generic and does not need to assume any specific algorithmic form or even any specific function class (like convex functions). We also prove that most existing families of online learning algorithms are in fact stable in our sense and using our connections provide simple regret bound analysis for them. Another related work [16]

considers an online algorithm, namely stochastic gradient descent (SGD) algorithm, in the iid setting where each function

is samples points in an iid fashion from some distribution. In this setting, [16] defines a new notion of online stability which is motivated by uniform stability [4]. The paper shows that SGD satisfies the new notion of stability and provides consistency guarantees as well. In contrast, our fundamental results connecting stability with regret hold for any algorithm and for any set of adversary moves , not just those sampled iid from a distribution.

A general class of online learning algorithms are referred to as Follow-The-Leader (FTL) [7] algorithms. At step , this algorithm chooses the element of which minimizes the sum of the functions played by the adversary up to that point:

 wt+1=argminw∈Ct∑i=1fi(w) . (1)

It can be shown that surprisingly simple algorithm achieves regret when the adversary is restricted to playing strongly convex functions [10].

A generalization of FTL is by adding a regularizer which results in the Follow-The-Regularized-Leader (FTRL) algorithm [17, 1]. In this case the update is given by

 wt+1=argminw∈Ct∑i=1ηfi(w)+R(w) (2)

Typically, is a strongly convex regularizer with respect to the appropriate norm and is a tradeoff parameter. Another way of describing FTRL algorithms is using Bregman divergences [17]. In particular, by defining and , we can write FTRL update in an equivalent form:

 wt+1=argminw∈Cηft(w)+Dϕt−1(~w)

where is the corresponding unconstrained minimizer.

Another class of algorithms is the proximal type algorithms also called Mirror Descent(MD) methods [15], that typically tries to find an iterate close to the previous iterate but also minimizes the current loss function and obtains same rates of regret as FTRL. Similar to FTRL, such algorithms also achieves regret for general convex functions and regret for strongly convex functions. It is interesting to note that Zinkevich’s algorithm [25] is just a special case of mirror descent with the Euclidean norm and and is similar to a stochastic gradient descent update [3].

While mirror descent and FTRL look fundamentally different algorithms and were considered to be two different ends of the spectrum for online learning algorithms [21], a recent paper [14] shows equivalence between different mirror descent algorithms and corresponding FTRL counterparts. In particular they show that the FOBOS mirror descent algorithm [9] is conceptually similar to Regularized Dual Averaging (RDA) [24] with minor differences emanating out of usage of proximal strongly convex regularizer and handling of arbitrary nonsmooth regularization like the norm. These difference result in different sparsity properties of the two algorithms.

5 Three conditions for online learning

In this section, we formally define our stability notion as well as introduce our bounded forward regret condition. We show that given stability, bounded regret and bounded forward regret are equivalent. Moreover, any algorithm with bounded regret can be converted into a stable algorithm with bounded regret and forward regret. Finally, we consider several existing OCP algorithms and illustrate that our forward regret and stability conditions can be used to provide a simple recipe for proving regret. For each of the algorithms, our novel analysis simplifies existing analysis significantly and in some cases also tightens the analysis.

We first define the following three quantities for any online learning algorithm:

• Online Stability: Intuitively, an online algorithm is defined to be stable if the consecutive iterates generated by are not too far away from each other. Formally, if is the point selected by at the -th step, then the (cumulative) online stability of is given by

 SA(T)=T∑t=1∥wt−wt+1∥. (3)

Now, if , then we say that is online stable. of stability is closely related to [20] (See Definition 17). Next, we define a stronger definition of stability, which we call Uniform Stability:

 USA(t)=∥wt−wt+1∥. (4)

If , then is defined to be uniformly stable. Clearly, if is uniformly stable then it is (cumulatively) stable as well. In section 6, we show that most of the existing online learning methods are actually uniformly stable. Interestingly, for COMiD (see section 7.4), while proving cumulative stability is relatively straightforward, one can show that uniform stability need not hold in general.

• Forward Regret: Forward regret is the hypothetical regret incurred by if it had access to the next move that the adversary was going to make. Note that forward regret cannot actually be attained by an algorithm since it depends on seeing one step into the future. Formally,

 FRA(T)=T∑t=1[ℓt(wt+1)−ℓt(w∗)], (5)

where . We define to have bounded (or vanishing) forward regret if . Note that if the online algorithms are randomized, we can replace the three quantities with their expected counterparts and all the bounds in the paper still hold.

• Regret: Regret is a standard notion in online learning that measures how good the steps of the algorithm are compared to the best fixed point in hindsight:

 (6)

Here again, if , then is said to have bounded (or vanishing) regret.

These three concepts, besides being important in their own right, are also intimately related. In particular, in the next section we show that given any two of these conditions, the third condition holds.

5.1 Connections between the three conditions

In this section, we show that the three conditions (i.e., bounded stability, bounded forward regret and bounded regret) defined in the previous section are closely related in the sense that given any two of the conditions, the third condition follows directly. For our claim, we first show that assuming stability,

 bounded forward regret⟺bounded regret .

We then prove that bounded regret can be shown to exhibit stability, albeit with worse rates of regret. Our claims are formalized in the following theorems.

Theorem 4.

Assume an online algorithm satisfies the condition of online stability (3) where the function played by the adversary at each step is -Lipschitz. Then, we have,

 RA(T)≤L⋅SA(T)+FRA(T), (7) FRA(T)≤L⋅SA(T)+RA(T).

Therefore, assuming online stability of , bounded forward regret and bounded regret are equivalent conditions.

Proof.

We first assume that has online stability and bounded forward regret. We have

 T∑t=1[ℓt(wt)−ℓt(w∗)] ≤T∑t=1L∥wt−wt+1∥+FR(T)≤L⋅S(T)+FR(T)=o(T),

where the second last inequality follows by Lipschitz continuity of and the last equality holds as both . Hence, has bounded regret. The proof in the reverse direction follows identically. ∎

To complete the picture regarding the connections between the three conditions, we now prove the following theorem.

Theorem 5.

Let be a fixed set of bounded diameter from which a learner selects a point at each step of online learning. Let be the class of -Lipschitz functions from which the adversary plays a function at each step. Also, let have bounded regret. Then, there exists a stable algorithm with bounded regret and forward regret.

Proof.

Intuitively, our proof proceeds by constructing an alternative stable algorithm that averages a batch of loss functions and feeds it into the “unstable” but bounded regret algorithm . We then show bounded regret and forward regret of this new algorithm. Note that our proof strategy is inspired by the proof of Lemma 20 in [22] that shows stability to be a necessary condition for learnability in batch setting.

Formally, given the algorithm , we construct a new algorithm in the following way. We divide the set of points into batches of and repeats the same point in an entire batch. At the end of the batch, it feeds the average of the functions in the batch to to get its next move. It now sticks to this new point for the next time steps before repeating the process all over. In picture,

 A′ sees:A% sees:ℓ1,ℓ2,…,ℓBB⋅g1,ℓB+1,…,ℓ2BB⋅g2,……

Note that the function , being an average of Lipschitz functions, is itself Lipschitz. Denote the elements generated by as and those by as . Note that there are only distinct elements in this sequence: viz. the elements generated by in response to . The stability analysis of now proceeds as follows

 T∑t=1∥∥w′t−w′t+1∥∥=⌊T/B⌋∑t=1∥∥w′(t−1)B+1−w′tB+1∥∥=⌊T/B⌋∑t=1∥wt−wt+1∥≤TBD=o(T),

for the choice in particular. This proves that is stable.

In order to show that has bounded regret, we consider

 T∑t=1(ℓt(w′t)−ℓt(w∗)) ≤B⌊T/B⌋∑t=1(ℓt(w′t)−ℓt(w∗))+L⋅D⋅B=⌊T/B⌋∑i=1iB∑t=(i−1)B+1(ℓt(w′t)−ℓt(w∗))+LDB =B⌊T/B⌋∑i=1(gi(wi)−gi(w∗))+L⋅D⋅B≤B⋅RA(⌊T/B⌋)+L⋅D⋅B,

where as has bounded regret. The last term in the first inequality is an upper bound on the regret due to the last batch of functions (maximally in number). Selecting , we get and hence the above bound is , i.e, has bounded regret. ∎

Thus we show that given any algorithm with bounded regret, we can convert it into another online stable algorithm with bounded regret which also implies bounded forward regret using Theorem 4.

6 Unified analysis of online algorithms

In this section we present examples where existing online learning algorithms can be analyzed through our stability and forward regret conditions and hence lead to regret bounds directly (see Theorem 4). These examples illustrate that the stability and forward regret conditions are critical to regret analysis and in fact provide a fairly straightforward recipe for regret analysis of online learning algorithms. Note that, unlike the general results of section 5, here we will make convexity assumptions on and . One of the major contributions of this paper is that our analysis significantly simplifies as well as tightens up analysis for existing methods like IOL [12].

Before delving into the technical detials, we provide a brief generic sketch of the regret analysis of all the algorithms.

For each of the regret analyses, initially we bound the stability in terms of the learning rate and the Lipschitz coefficient of , . The bounds on stability are generally obtained by exploiting the optimality of at iteration , the lipschitz continuity of and the strong convexity of the regularizer (for the algorithms involving regularization). For the case of IOL, , which makes the stability bounded by .

For FTL, forward regret is non positive by definition of the FTL updates. For all the other algorithms, the bounds on the forward regret follow by again using the optimality of at iteration and comparing the corresponding objective at the final minimizer . This generally results in a telescoping sum, upper bounding the forward regret in terms of the regularizer (or the bregman divergence ) evaluated at the extreme iterates and with all the other terms canceling out by appropriately choosing . In particular, for the case of IOL, the forward regret is bounded by .

Finally bounds on the regret are obtained by using equation (7) while the optimum dependence on are obtained by trading off the step size in the corresponding inequality. Summation over appropriate gives us rates of regret for strongly convex and rates of regret for general convex lipschitz as is common in the literature.

7 Examples

Follow the leader(FTL) is a popular method for OCP when the provided functions are strongly convex. At the -th step FTL chooses to be the element that minimizes the total loss up to that step, i.e.,

 FTL:wt+1=argminw∈Ct∑τ=1ℓτ(w). (8)

The FTL method was analyzed in [7] and [21] for the case when each loss function is at least -strongly convex. Here, using our forward regret and stability conditions, we provide a significantly simpler analysis with similar regret bounds. It should be noted that our analysis is a generalization of the analysis in [7, Section 3.2] from strongly convex functions w.r.t. norm to strongly convex functions w.r.t. arbitrary norm.

Theorem 6.

Let each loss function be -strongly convex and -Lipschitz continuous. Then, the regret incurred by FTL algorithm (see (8)) is bounded by:

 RFTL(T)≤2L2α(1+lnT).
Proof.

Our proof follows the simple recipe of computing stability as well as forward regret bound.

Stability: Using strong convexity, Lemma 3 and the fact is the optimum of (8),

 t∑τ=1ℓτ(wt)≥t∑τ=1ℓτ(wt+1)+tα2∥wt−wt+1∥2. (9)

Similarly, using optimality of for the -th step:

 t−1∑τ=1ℓτ(wt+1)≥t−1∑τ=1ℓτ(wt)+(t−1)α2∥wt−wt+1∥2. (10)

Adding (9) and (10), and by using Lipschitz continuity of we get:

 ℓt(wt)−ℓt(wt+1) ≥(t−1/2)α∥wt−wt+1∥2, ⟹L(t−1/2)α ≥∥wt−wt+1∥. (11)

Using (11), we get:

 T∑t=1∥wt−wt+1∥ ≤T∑t=12L(2t−1)α≤2Lα(1+lnT). (12)

Hence,

 SFTL(T)≤2Lα(1+lnT). (13)

Forward Regret: Using optimality of for -th step:

 T∑t=1ℓt(w∗)≥T∑t=1ℓt(wT+1). (14)

Next using (10) for and (14),

 T∑t=1ℓt(w∗)≥ℓT(wT+1)+T−1∑τ=1ℓτ(wT). (15)

Similarly using (10) with (15) for ,

 T∑t=1ℓt(w∗)≥T∑t=1ℓt(wt+1). (16)

Hence,

 FRFTL(T)=0. (17)

Hence, using Theorem 4, (13), and (17),

 RFTL(T)≤2L2α(1+lnT). (18)

While FTL is an intuitive algorithm, unfortunately, for non-strongly convex functions it need not have bounded regret. However, several recent results show that by adding strongly convex regularization, FTL can be used to obtain bounded regret. Specifically,

 FTRL:wt+1=argminw∈Ct∑τ=1ℓτ(w)+1ηR(w). (19)

where is (generally) a strongly convex function with respect to an appropriate norm. Note that the intuition behind adding a regularization is making the algorithm stable. Our analysis of FTRL explicitly captures this intuition by showing the existence of stability condition, while forward regret follows easily from the forward regret analysis of FTL given above.

Theorem 7.

Let each loss function be -Lipschitz continuous, diameter (as measured in ) of set be , and let be a -strongly convex regularization function. Then, the regret incurred by Follow The Regularized Leader (FTRL) algorithm (see (19)) is bounded by:

 RFTRL(T)≤2L√∥∇R∥∗D√T ,

where .

Proof.

As for FTL, we again prove regret by first proving stability and forward regret.
Stability: Similar to (9) and (10), using strong convexity and optimality conditions for -th and -th step, we get the following relations:

 t∑τ=1ℓτ(wt)+1ηR(wt) ≥t∑τ=1ℓτ(wt+1)+1ηR(wt+1)+12η∥wt−wt+1∥2. (20) t−1∑τ=1ℓτ(wt+1)+1ηR(wt+1) ≥t−1∑τ=1ℓτ(wt)+1ηR(wt)+12η∥wt−wt+1∥2. (21)

Combining (7.2) and (7.2) and by Lipschitz continuity of :

 Lη≥∥wt−wt+1∥. (22)

Hence,

 SFTRL(T)=T∑t=1∥wt−wt+1∥≤LηT. (23)

Choosing satisfies the online stability condition of FTRL.

Forward Regret: Assuming and , FTRL is same as FTL with an additional -th step loss function . Hence using (16), we obtain:

 T∑t=1ℓt(w∗)+1η(R(w∗)−R(w1))≥T∑t=1ℓt(wt+1). (24)

Hence,

 FRFTRL(T)=1η(R(w∗)−R(w1))≤∇R(w∗)⊤(w∗−w1)η≤∥∇R∥∗Dη. (25)

where the first inequality follows using the convexity of and the last one follows using Cauchy Schwartz inequality. Again provides vanishing forward regret for FTRL. Hence, using Theorem 4,

 RFTRL(T)≤∥∇R∥∗Dη+L2ηT≤2L√∥∇R∥∗D√T. (26)

by appropriately choosing to be . ∎

7.3 Regularized Dual Averaging (RDA)

Regularized Dual Averaging [24] is a popular online learning method to handle OCP scenarios where each loss function is regularized by the same regularization function, i.e., functions at each step are of the form , where is a regularization function. RDA computes the iterates using following rule:

 (27)

where , is a strongly convex regularizer that is separately added and is the trade-off parameter. [24] shows that the above update obtains regret for general Lipschitz continuous functions and regret when the regularizer is strongly convex.

Note that RDA is same as FTRL except for linearization of the first part of loss function . Hence, same regret analysis as FTRL should hold. However, analysis by [24] shows that by using special structure of , regret can be bounded even without assuming Lipschitz continuity of the regularization function . Below, we show that using the same recipe of bounding stability and forward regret leads to significantly simpler analysis of RDA as well. Unlike the previous cases, this analysis is slightly more tricky as we cannot assume Lipschitz continuity of to prove stability.

Theorem 8.

Let each loss function be -Lipschitz continuous, be a -strongly convex function and wlog . Now, using at each step, regret of RDA (see (27)) is bounded by .

Proof.

Stability: By strong convexity of and optimality of and for the -th and -th step respectively,

 1tt∑τ=1g⊤τ(wt−wt+1)+r(wt)−r(wt+1) ≥α2∥wt−wt+1∥2, 1t−1t−1∑τ=1g⊤τ(wt+1−wt)+r(wt+1)−r(wt) ≥α2∥wt−wt+1∥2.

 α∥wt−wt+1∥2≤(1tgt−1t(t−1)t−1∑τ=1gτ)⊤(wt−wt+1)≤2Lt∥wt−wt+1∥, (28)

where the second inequality follows from Lipschitz continuity of . After simplification and adding the above expression for all ,

 SRDA(T)≤2Lα(1+lnT). (29)

Note that the above stability analysis is slightly different from that of FTL as we are able to bound the stability by Lipschitz constant of only, rather than .

Forward Regret: When , forward regret follows easily from forward regret of FTL where loss function at each step is . Hence,

 FRRDA(T)≤0. (30)

Hence, using Theorem 4,

 T∑t=1(g⊤t(wt−w∗)+r(wt)−r(w∗))≤2L2α(1+lnT).

The result now follows using convexity of , i.e., . ∎

Next, we bound regret incurred by RDA for general convex, Lipschitz continuous functions.

Theorem 9.

Let each loss function be -Lipschitz continuous and wlog and , . Now, using at each step, regret of RDA (see (27)) is bounded by .

Proof.

Stability: Again, by strong convexity of and optimality of and for the -th and -th step respectively,

 1tt∑τ=1gτ⋅(wt−wt+1)+r(wt)−r(wt+1)+βtt(h(wt)−h(wt+1))≥βt2t∥wt−wt+1∥2,
 −15pt1t−1t−1∑τ=1gτ⋅(wt+1−wt)+r(wt+1)−r(wt)+βt−1t−1(h(wt+1)−h(wt))≥βt−12(t−1)∥wt−wt+1∥2.

Adding the above two equations, using Lipschitz continuity of and upper bound on ,

 (12√t+12√t−1)∥wt−wt+1∥2−2Lt∥wt−wt+1∥−(1√t−1−1√t)D2≤0 (31)

Solving for , we get,

 ∥wt−wt+1∥≤2L+D√t−1. (32)

Hence,

 SRDA(T)≤(2L+D)√T. (33)

Forward Regret: Using optimality of ,

 T∑t=1g⊤tw∗+Tr(w∗)+√Th(w∗)≥T∑t=1gt⋅wT+1+Tr(wT+1)+√Th(wT+1). (34)

Now, using optimality of ,

 T−1∑t=1g⊤twT+1+(T−1)r(wT+1)+√T−1h(wT+1)≥T−1∑t=1gt⋅wT+(T−1)r(wT)+√T−1h(wT). (35)

 T∑t=1g⊤tw∗+Tr(w∗)+√Th(w∗)≥g⊤TwT+1+r(wT+1)+T−1∑t=1g⊤twT+(T−1)r(wT)+√T−1h(wT). (36)

Similarly, combining optimality of in (35) recursively with (36),

 T∑t=1g⊤tw∗+Tr(w∗)+√Th(w∗)≥T∑t=1(g⊤twt+1+r(wt+1)). (37)

Hence, using and ,

 FRRDA(T)≤√TD2. (38)

Hence, using Theorem 4 and convexity of each ,

 RRDA(T)≤(D2+L(2L+D))√T. (39)

7.4 Composite Objective Mirror Descent (COMiD)

Similar to RDA, COMiD [8] is also designed to handle regularized loss functions of the form . Just as RDA is an extension of FTRL to handle composite regularized loss functions, similarly, COMiD is an extension of IOL. Formally,

 COMiD:wt+1=argminw∈Cη(g⊤tw+r(w))+DR(w,wt),

where , is the Bregman divergence with being the generating function. Now, similar to RDA, regret analysis of COMiD follows directly from regret analysis of IOL. However, [8] presents an improved analysis, that can handle non-Lipschitz continuous regularization as well. Here, we show that using our stability/forward-regret based recipe, we can also obtain similar regret bounds with significantly simpler analysis.

Theorem 10.

Let each loss function be of the form , where is a -Lipschitz continuous function and is a regularization function. Let diameter of set be , and let be a Bregman divergence with being the convex generating function. Let . Also, let be a positive function. Then, the regret incurred by the Composite Objective Mirror Descent (COMiD) algorithm is bounded by:

 RCOMiD(T)≤L√2R(w∗)√T.

Furthermore, if each function is -strongly convex w.r.t. , then

 RCOMiD(T)≤2L2α(1+lnT)+αR(w∗).
Proof.

Stability: By optimality of ,

 ηt(gt⋅wt+r(wt))≥DR(wt+1,wt)+ηt(gt⋅wt+1+r(wt+1)), (40)

Adding the above inequality for and using the fact that (by the definition of ),

 T∑t=112Lηt∥wt−wt+1∥2≤T∑t=1∥wt−wt+1∥. (41)

Using Cauchy-Schwarz inequality,

 (T∑t=11√2Lηt√2Lηt∥wt−wt+1∥)2≤T∑t=112Lηt∥wt−wt+1∥2T∑t=12Lηt. (42)

Using (41) and (40),

 SCOMiD(T)=T∑t=1∥wt−wt+1∥≤2LT∑t=1ηt. (43)

Forward Regret: Forward regret follows directly from the forward regret of IOL (53), i.e,

 FRCOMiD =T∑t=1(g⊤t(wt+1−w∗)+r(wt+1)−r(w∗)) (44) (45)

Both the regret bounds follow using convexity of each and setting step sizes as in IOL (see (54), (55)). ∎

7.5 Mirror Descent (MD)

Mirror descent algorithms are a generalization of Zinkevich’s Gradient Infinitesimal Gradient Ascent (GIGA) algorithms [25] where regularization can be drawn from any Bregman distance family. Formally,

 (46)

where is the Bregman divergence generated using . Note that MD update is the same as COMiD with . Hence, our stability analysis as well as regret analysis for general convex functions follows directly. However, for strongly convex functions, our approach does not yield appropriate forward regret directly; primary reason being linearization of the function. Instead, we can obtain regret bound using standard approach (see [25]) and then obtain forward regret bound using Theorem 4.

7.6 Implicit Online Learning (IOL)

Implicit online learning [12] is similar to typical Mirror Descent algorithms but without linearizing the loss function. Specifically at iteration ,

 IOL:wt+1=argminw∈C(DR(w,wt)+ηtℓt(w)), (47)

where is a Bregman’s divergence with being the generating function. It was shown in [12] that using any strongly convex , the above update leads to regret for any Lipschitz continuous convex functions . This paper also shows that if is selected to be squared -norm and each function is strongly-convex and has Lipschitz continuous gradient, then regret can also be achieved. Below, using our recipe of forward regret and stability we reproduce significantly simpler proofs for both as well as regret. Furthermore, our proof requires only strong-convexity and Lipschitz continuity, in contrast to strong-convexity and Lipschitz continuity of the gradient in [12]. Also, our analysis can handle any strongly convex , rather than just the squared -norm regularizer.

Theorem 11.

Let each loss function be -Lipschitz continuous, diameter of set be , and let be a Bregman divergence with being the strongly convex generating function. Also, let be a positive function. Then, the regret incurred by the Implicit Online Learning (IOL) algorithm (see (47)) is bounded by:

 RIOL(T)≤2L√2R(w∗)√T.

Furthermore, if each function is -strongly convex w.r.t i.e. , , then

Proof.

Here again, we follow the recipe of proving stability and forward regret.
Stability: Stability again follows easily by using optimality of and comparing it to . Formally,

 ηtℓt(wt) ≥DR(wt+1,wt)+ηtℓt(wt+1), ηtℓt(wt) ≥12∥wt+1−wt∥2+ηtℓt(wt+1), 2Lηt <


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Session 87 - Environments of QSOs and Radio Galaxies.
Oral session, Wednesday, January 17
Corte Real, Hilton

## [87.04] HST Imaging of the Host Galaxies of Quasars Selected by Radio and Optical Properties

E. J. Hooper (Steward Observatory)

A sample of 16 quasars from the Large Bright Quasar Survey (LBQS) has been imaged with WFPC2 on the Hubble Space Telescope. Host galaxies are clearly visible around several of the quasars. Many fields contain companions with small projected separations from the host galaxies, and there is evidence of interaction between the companion and the host galaxy in some cases. The sample was selected to include radio-loud and radio-quiet quasars in a narrow range of redshift (0.4 \leq z \geq 0.5) at faint to moderate absolute magnitudes (-23 < M_B < -25).

Radio observations of the LBQS indicate that the radio-loud fraction of optically selected quasars is mostly independent of absolute magnitude and redshift (Hooper et al. 1995, ApJ, 445, 62; Hooper et al. 1995, ApJ, submitted). No correlation is detectable between radio luminosity and absolute magnitude among radio-loud quasars, and the radio-loud fraction remains constant at \approx 10% from M_B = -23 to M_B \geq -28. In particular, no decrease in radio-loud fraction for M_B fainter than -24 is seen, contrary to several previous studies based on the Palomar-Green quasar survey (e.g., Peacock et al. 1986, MNRAS, 218, 265). The radio-loud fraction does not evolve, aside from a modest peak at z \sim 1, from z = 0.2 to redshifts approaching 5, based on the LBQS and three high-redshift quasar samples.


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>> Home >> Resources & support >> FAQs >> Dialog programming (part 2)—Adding features

### How do I add features to Stata dialogs?

 Title Dialog programming (part 2)—Adding features Authors James Hassell, StataCorp Jean Marie Linhart, StataCorp Date August 2003; minor revisions July 2011

This FAQ is intended to be a supplement for the documentation rather than a substitute. We strongly recommend reading the online help on dialog programming, which contains information that will not be covered here.

In the previous section, Dialog programming (part 1)—Getting started, we created a simple dialog for Stata's generate command. This dialog was called mygenerate, and it was saved as mygenerate.dlg. Our first mygenerate dialog only contained two input controls. These controls allowed the user to create a new variable and to define the contents of that variable. Stata's generate command has other features, too. It has options for setting the type of the new variable and for attaching a value label. Furthermore, it accepts if and in arguments. Wouldn't it be nice if our mygenerate dialog had these features?

This FAQ is going to deal specifically with adding the features mentioned above to our existing mygenerate dialog. The type option and the lblname option for Stata's generate command have one similarity: they offer choices to the user. For example, the type option can include any of the following arguments:

        float
double
int
long
byte
str

Likewise, the lblname option generally accepts labels that have already been defined. Therefore, the best presentation for both options will be to offer the user a list of valid choices. A COMBOBOX control is the natural choice.

As stated above, the type and lblname options are similar and are both best presented as a COMBOBOX. That, however, is where the similarity ends. Specifically, the COMBOBOX associated with the type option will be populated with a static list that we must define. Alternatively, the COMBOBOX for the lblname option will need to reflect the value labels that are defined for the current dataset.

The code segment below reflects the additions discussed above for mygenerate.dlg.

Once you have downloaded it, save mygenerate.dlg in your Stata adopath, most preferably your PERSONAL directory. If you are unfamiliar with the PERSONAL directory, please see the following FAQ: Where is my personal ado directory. If you downloaded the previous version listed in Dialog Programming (part 1)—Getting started, you can overwrite it.

Once it has been saved correctly, mygenerate can be called from the Stata command line by typing db mygenerate.

A side note:
It is always a good idea to use the discard command after you make changes to dialog code. Using the discard command before loading the new version of a dialog will clear the memory in the class system and prevent Stata from entering an unstable state.

------------------------- mygenerate.dlg -------------------------
VERSION 12

POSITION . . 410 250

DIALOG main, label("generate - Generate a new variable") tabtitle("Main")
BEGIN
TEXT     tx_gen      10  10   120  ., label("Generate variable:")
EDIT     ed_gen      10  +20  120  ., error("Generate variable")
TEXT     tx_exp      10  +25  390  ., label("Contents:")
EXP      ex_exp      10  +20  390  ., label("Create ...")        ///
error("Contents")

TEXT     tx_type     10  +30  200  .,                            ///
label("Generate variable as type:")
COMBOBOX cb_type     @   +20  120  .,                            ///
dropdownlist                                            ///
contents(type_list)

CHECKBOX ck_vlab     215 -20  185  .,                            ///
label("Attach value label:")                            ///
onclickon(script main_ck_vlab_on)                       ///
onclickoff(script main_ck_vlab_off)
COMBOBOX cb_vlab     @   +20  @    .,                            ///
contents(valuelabels) dropdown

END

LIST type_list
BEGIN
float
double
long
int
byte
str
END

SCRIPT main_ck_vlab_on
BEGIN
main.cb_vlab.enable
END

SCRIPT main_ck_vlab_off
BEGIN
main.cb_vlab.disable
END

OK ok1,      label("OK")
CANCEL can1, label("Cancel")
SUBMIT sub1, label("Submit")
HELP hlp1,   view("help generate")
RESET res1
COPY copy1

PROGRAM command
BEGIN
put "generate "
put main.cb_type " "
require main.ed_gen
put main.ed_gen " "
if main.ck_vlab {
put ": "
put main.cb_vlab " "
}
put "= "
require main.ex_exp
put main.ex_exp
END
------------------------- mygenerate.dlg -------------------------

The code above produces the following screen image.

The first change made to the code above was the addition of the new controls to the main DIALOG. Notice that four new controls were added. They are a TEXT control, two COMBOBOX controls, and one CHECKBOX control. The positions were defined for these controls much like they were in the original version of mygenerate. The one notable exception is the use of the "@" symbol, which simply instructs the current dialog control to inherit the respective position or size attribute from the previously defined control. The two examples below have identical meaning.

        Example 1:
TEXT     tx_one     10  30   200  ., label("Label one")
TEXT     tx_two     10  +30  200  ., label("Label two")

Example 2:
TEXT     tx_one     10  30   200  ., label("Label one")
TEXT     tx_two     @   +30  @    ., label("Label two")

We will analyze the new dialog code line by line. An explanation for the new TEXT control will be omitted, as that topic was discussed in Dialog programming (part1)—Getting started.

        -  COMBOBOX cb_type     @   +20  120  .,              ///
dropdownlist                              ///
contents(type_list)

By now, you should be familiar with the positioning of controls, so further explanation will not be given here. The dropdownlist option defines the type of COMBOBOX. Here, the COMBOBOX drops down when the user clicks the control, and it contains a list that cannot be edited. The contents option specifies the name of the LIST that contains the entries to be displayed. Here, the LIST is defined below this DIALOG section and is named type _list.

        -  CHECKBOX ck_vlab     215 -20  185  .,              ///
label("Attach value label:")               ///
onclickon(script main_ck_vlab_on)          ///
onclickoff(script main_ck_vlab_off)

The CHECKBOX control above has two options to handle events associated with clicking the CHECKBOX on or off: onclickon and onclickoff. This type of event handling is useful because it allows the dialog programmer to control precisely how controls interact. Below this DIALOG section, two SCRIPTS have been defined that perform specific tasks. These SCRIPTS will be discussed in more detail below. In the context of this CHECKBOX, the most important issue is that we call a specific SCRIPT when the user clicks this box on or off.

        -  COMBOBOX cb_vlab     @   +20  @    .,              ///
contents(valuelabels) dropdown

This COMBOBOX control is similar to the one defined above, except that it uses the valuelabels list, which is defined within Stata. The valuelabels list will always contain the value labels that are defined for the current dataset. Another difference between this COMBOBOX and the one defined above is that the one above is a dropdownlist, and this one is a dropdown. These two types of controls are very similar, the only real difference being that the dropdown option will allow users to type their own entry into the list.

The discussion above mentioned two new constructs, LIST and SCRIPT. In our example, both of these were added just below the DIALOG definition, but they could have just as easily been added above it. The placement is purely a matter of style and does not affect functionality.

        LIST type_list
BEGIN
float
double
long
int
byte
str
END

This section defines a LIST that can be used to populate a COMBOBOX or LISTBOX. Simply include the name of the LIST in the control's contents option.

        SCRIPT main_ck_vlab_on
BEGIN
main.cb_vlab.enable
END

SCRIPT main_ck_vlab_off
BEGIN
main.cb_vlab.disable
END

The section above contains two SCRIPT definitions. A SCRIPT is used to define a compound i-action or sequence of i-actions. The SCRIPT definitions in this example include only a single i-action, but they could have just as easily listed a sequence of i-actions, if necessary. Recall that these scripts were invoked by the onclickon and onclickoff options associated with a CHECKBOX.

        PROGRAM command
BEGIN
put "generate "
put main.cb_type " "
require main.ed_gen
put main.ed_gen " "
if main.ck_vlab {
put ": "
put main.cb_vlab " "
}
put "= "
require main.ex_exp
put main.ex_exp
END

The section above is very similar to the original version of mygenerate, except that code has been added to handle the syntax of the new features. These lines will be addressed in sequence below.

        -    put main.cb_type " "

This line adds the type of the new variable and a space to the return string. The type that is added depends on the cb_type that is selected from the main tab of the COMBOBOX.

        -   if main.ck_vlab {

This line is the beginning of a conditional statement. The statement means that when main.ck_vlab is selected (checked), the program will execute the enclosed block of code; otherwise, it will skip the enclosed block of code.

        -   put ": "

The code adds a colon followed by a space to the return string. This syntax is necessary for Stata's generate command when the lblname is specified.

        -   put main.cb_vlab " "

Next the code adds the contents of main.cb_vlab followed by a space to the return string. This specifies the lblname.

        -   }

The right brace ends the conditional block.

This section started by mentioning some of the features that were missing in the mygenerate dialog. We have now successfully added some of those features, which include the ability to specify the data type and the lblname. However, we have not added the ability to specify if and in. The next section will focus on adding these as a new dialog tab.


text:
MathOverflow will be down for maintenance for approximately 3 hours, starting Monday evening (06/24/2013) at approximately 9:00 PM Eastern time (UTC-4).

2 added 3 characters in body

As Harry suggests in his answer, it is probably more intuitive to work with associated primes, rather than the slightly older language of primary decompositions.

If $I$ is an ideal in $A$, an associated prime of $A/I$ is a prime ideal of $A$ which is the full annihilator in $A$ of some element of $A/I$. A key fact is that for any element $x$ of $A/I$, the annihilator of $x$ in $A$ is contained in an associated prime.

The associated primes are precisely the primes that contribute to the primary decomposition of $I$. Geometrically, $\wp$ is an associated prime of $A/I$ if there is a section of the structure sheaf of Spec $A/I$ that is supported on the irreducible closed set $V(\wp)$. E.g. in the example given in Cam's answer, the function $x^2 - x$ is not identically zeor zero on $X:=$ Spec ${\mathbb C}[x,y]/(x y, x^3-x^2, x^2 y - xy),$ but it is annihilated by $(x,y)$, and so is supported at the origin (if we restrict it to the complement of $(0,0)$ in $X$ then it becomes zero).

The non-minimal primes of $I$ that play a role in the primary decomposition of $I$ (i.e. appear as associated primes of $A/I$) are the generic points of the so-called embedded components of Spec $A/I$: they are irreducible closed subset of Spec $A/I$ that are not irreducible components, but which are the support of certain sections of the structure sheaf.

An important point is that if $I$ is radical, so that $A/I$ is reduced, then there are no embedded components: the only associated primes are the minimal primes (for the primary decomposition of $I$ is then very simple, as noted in the question: $I$ is just the intersection of its minimal primes).

There is a nice criterion for a Noetherian ring to be reduced: Noetherian $A$ is reduced if and only if $A$ satisfies $R_0$ and $S_1$, i.e. is generically reduced, and has no non-minimal associated primes. GeoemtricallyGeometrically, and applied to $A/I$ rather than $A$, this says that if $A/I$ is generically reduced, then the embedded components are precisely the irreducible closed subsets of Spec $A/I$ over which the nilpotent sections of the structure sheaf are supported. This may help with your nilpotentification'' mental image.

1

As Harry suggests in his answer, it is probably more intuitive to work with associated primes, rather than the slightly older language of primary decompositions.

If $I$ is an ideal in $A$, an associated prime of $A/I$ is a prime ideal of $A$ which is the full annihilator in $A$ of some element of $A/I$. A key fact is that for any element $x$ of $A/I$, the annihilator of $x$ in $A$ is contained in an associated prime.

The associated primes are precisely the primes that contribute to the primary decomposition of $I$. Geometrically, $\wp$ is an associated prime of $A/I$ if there is a section of the structure sheaf of Spec $A/I$ that is supported on the irreducible closed set $V(\wp)$. E.g. in the example given in Cam's answer, the function $x^2 - x$ not identically zeor on $X:=$ Spec ${\mathbb C}[x,y]/(x y, x^3-x^2, x^2 y - xy),$ but it is annihilated by $(x,y)$, and so is supported at the origin (if we restrict it to the complement of $(0,0)$ in $X$ then it becomes zero).

The non-minimal primes of $I$ that play a role in the primary decomposition of $I$ (i.e. appear as associated primes of $A/I$) are the generic points of the so-called embedded components of Spec $A/I$: they are irreducible closed subset of Spec $A/I$ that are not irreducible components, but which are the support of certain sections of the structure sheaf.

An important point is that if $I$ is radical, so that $A/I$ is reduced, then there are no embedded components: the only associated primes are the minimal primes (for the primary decomposition of $I$ is then very simple, as noted in the question: $I$ is just the intersection of its minimal primes).

There is a nice criterion for a Noetherian ring to be reduced: Noetherian $A$ is reduced if and only if $A$ satisfies $R_0$ and $S_1$, i.e. is generically reduced, and has no non-minimal associated primes. Geoemtrically, and applied to $A/I$ rather than $A$, this says that if $A/I$ is generically reduced, then the embedded components are precisely the irreducible closed subsets of Spec $A/I$ over which the nilpotent sections of the structure sheaf are supported. This may help with your nilpotentification'' mental image.


text:
Browse Questions

# A source of sound producing wavelength 50 cm is moving away from a stationary observer with $\large\frac{1}{5}th$ speed of sound. Then what is the wavelength of sound heard by the observer?

$\begin {array} {1 1} (1)\;55\: cm & \quad (2)\;40\: cm \\ (3)\;60\: cm & \quad (4)\;70\: cm \end {array}$

Can you answer this question?

(3) 60 cm
answered Nov 7, 2013 by


text:
Where the income statement can be compared with previous years, and the net income can be compared where it helps to compare and understand the percentage of rising or loss of income percentage. In the vertical analysis of financial statements, the percentage is calculated by using the below formula: Vertical Analysis formula = Individual Item / Base Amount *100, Vertical analysis formula for the Income Statement and Balance Sheet are given below –. Excel's INDEX function allows users to reference values in a range of data (or array of data) by their column and row number position within that range. Vertical axis: Weibull cumulative probability expressed as a percentage Horizontal axis: ordered failure times (in a LOG10 scale) The vertical scale is ln(-ln(1-p)) where p=(i-0.3)/(n+0.4) and i is the rank of the observation. Example of the vertical analysis of the financial statement, which shows the total in amount and percentage. Cuntry song I. Karaoked. 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Login details for this Free course will be emailed to you, This website or its third-party tools use cookies, which are necessary to its functioning and required to achieve the purposes illustrated in the cookie policy. Sintaxis. Examples. You may learn more about financial statement analysis from the following articles –, Copyright © 2021. 1 Home > Formulas > Physics Formulas > Maximum Height Formula . Cuando bebé demuestre señales de hambre puedes hacer cosquillas cerca de la boca/mejilla con la mamadera (o método que usas) para que bebé abra bien su … Average velocity/speed of a moving object can be calculated as. Maximum Height Formula . geodetic positions in three dimensions. Please enable Cookies and reload the page. Clearly, in this example, the angle between the crank and the rod is not a right angle. Moreover, it would travel before it reaches the same vertical position as it started from. Amplitude, Period, Phase Shift and Frequency. But this method is not useful to make firm decisions, and measurement of the company value cannot be defined. EQUIV renvoie la position de la valeur cherchée dans le tableau. Vertical Analysis Formula = Individual Item / Total Assets (Liabilities) * 100. 4. AJ Design ☰ Math Geometry Physics Force Fluid Mechanics Finance Loan Calculator. Grade 10. But this method is not useful to make firm decisions, and measurement of the company value cannot be defined. eur-lex.europa.eu. In the previous section we motivated for using the same equations of motion as in rectilinear motion. method is one of the easiest methods of analyzing the financial statement. Typically, the MATCH function is used to find the position, which is then fed into INDEX… ground) or a minus sign (depth below the ground) using the format of two digits and two decimals. MATCH locates and feeds a position to INDEX,which returns the value at that position. Vertical analysis is also instrumental in comparing. Some functions (like Sine and Cosine) repeat forever and are called Periodic Functions.. Watch Queue Queue Historical definition. Mécanique Physique (S2) 2ème partie – page 3 pour remonter à la position if faudra « intégrer » (2 fois) les équations du mouvement, opération dont vous savez qu’elle est en général plu Or, put differently - a 'vlookup' returns a value in a specified column. Vertical distance. The vertical analysis of financial statements does not help to make a firm decision as there is no standard percentage or ratio regarding the change in the components of the income statement or the balance sheet. h = -16t 2 … While there are several sub-types of potential energy, we will focus on gravitational potential energy. Now I want to position the red box vertically in the parent div, but it must be flexible, because there are options that might extend the red box with one or two more text-rows. If the index mirror en el eje horizontal y un m.r.u.a. x 0 = 30 m. α = 3 m/s 2. t = 6s • Los bordes de la muestra deben tener cortes perpendiculares. It is much appreciated. CFA® And Chartered Financial Analyst® Are Registered Trademarks Owned By CFA Institute.Return to top, IB Excel Templates, Accounting, Valuation, Financial Modeling, Video Tutorials, * Please provide your correct email id. the initial position of the body is x 0. We can describe vertical projectile motion through a series of graphs: position, velocity and acceleration versus time graphs. Answer: Known : v 0 = 4m/s. Mouvement vertical de projectile : déterminer la vitesse initiale connaissant la durée totale. Topics. Where it helps to understand the percentage/share of the individual items; Where it helps to understand the structural composition of the various components like cost, expenses, assets, and liabilities. En estos movimientos la aceleración es la aceleración de la gravedad, g = 9,8m/s 2. Different organization statements can be compared as the comparison is made in percentage. European Journal of Applied Physiology, 38 (3): 181-188. Projectile Motion Equations Calculator Science Physics Formulas. 1. I.e., elevation would be the altitude of the ground or a building. Now from the Law of conversation of energy, we … position du mouvement des corps et les lois de la Mécanique qui font intervenir les dérivées secondes de la position. Practica encontrar la distancia entre dos puntos en el plano coordenado que comparten la misma coordenada x o la misma coordenada y. 2) Harman, E.A., Rosenstein, M.T., Frykman, P.N., & Rosenstein, R.M. The maximum height of the projectile depends on the initial velocity v 0, the launch angle θ, and … The same can be done like the income statement where the previous years can be compared and find out the change in the working capital and fixed assets over time. Where the altered business that requires a different amount on the ongoing fund. • Thevertical position(height or depth) [...] of the instruments shall be indicated in metres with a plus (= height above the. Tengo el siguiente problema: Requiero poder arrastrar una fórmula verticalmente, en la que hago una operación en … The formulas below show how INDEX can be used to get a value: = INDEX (A1:B5, 2, 2) // returns value in B2 = INDEX (A1:B5, 3, 1) // returns value in A3. Comentarios Remarks. Position with respect to crank angle (from the triangle relation, completing the square, ... Then, using the triangle law of sines, it is found that the rod-vertical angle is 18.60647° and the crank-rod angle is 88.21738°. Horizontal velocity. Excel's COLUMN function returns the number of the column in a referenced cell. Or we can measure the height from highest to lowest points and divide that by 2. Completing the CAPTCHA proves you are a human and gives you temporary access to the web property. Word Problem Exercises: Vertical Motion: Question Group #1: Directions and/or Common Information: The formula to model the height of an object t seconds after it has been dropped is . For a control placed directly on a screen, Parent refers to the screen. Vertical analysis of financial statement provides a comparable percentage which can be used to compare with the previous years. Segmental contribution to forces in vertical jump. If you are at an office or shared network, you can ask the network administrator to run a scan across the network looking for misconfigured or infected devices. Whereas the opposite of the vertical analysis of financial statements is the Horizontal analysis always looks at the amount from the financial statement over the horizon of many years. Quality analysis is not done by using vertical analysis of financial statements as there is no consistency in the ratio of the elements. If you're just given a problem and you were told a three kilogram mass is hanging from a vertical spring of spring constant 50 newtons per meter and this line here represents the equilibrium position, it's just hanging out right there at rest. The VLOOKUP function is a built-in function in Excel that is categorized as a Lookup/Reference Function.It can be used as a worksheet function (WS) in Excel. Salary paid to the workers of the company is $300000 office rent paid is$30000, utilities worth $40000 and other expenses are$60000. Convert vertical list to horizontal or vice versa with Transpose method. Description. Calculate the position of the person at the end time 6s if the initial velocity of the person is 4m/s and angular acceleration is 3 m/s 2. Where the total sales of company A are $1000000 and the cost of goods sold is$400000. You pull this mass down from the point where it was hanging at rest, 0.3 meter and let go. Transcription de la vidéo. Con una mano sostendrás su cabeza y cuello. If you are on a personal connection, like at home, you can run an anti-virus scan on your device to make sure it is not infected with malware. Physical Sciences; Grade 12; Vertical Projectile Motion in 1D; View Topics. The effects of arms and countermovement on vertical jumping. Vertical analysis formula = (Statement line item / Total base figure) X 100 Horizontal analysis formula = {(Comparison year amount - Base year amount) / Base year amount} X 100 However, it is important to remember that you can still use vertical analysis to compare a line item's percentages from one quarter or year to another. (b) Mirror horizon and real horizon form a single straight line — index arm is properly positioned. Vertical position = (horizontal position) (tangent of launch angle) – $$\frac{(acceleration due to gravity) (horizontal position)^{2}}{2(initial velocity)^{2}(cosine of launch angle)^{2}}$$ Approximate position Ecliptic coordinates. Hola, de antemano muchas gracias por su ayuda. Formulario de referencia Descripción. So by using this method, it is easy to understand the net profit as it is easy to compare between the years. Exercices : Exercices sur les MRUA . This method compares different items to a single item in the same accounting period. The maximum height of the object is the highest vertical position along its trajectory. Vertical position is not locked. Use TRANSPONER para cambiar la orientación vertical y horizontal de una matriz o un rango de una hoja de cálculo. Mouvement vertical de projectile : déterminer la hauteur maximale connaissant la durée totale. All Together Now! The VLOOKUP function performs a vertical lookup by searching for a value in the first column of a table and returning the value in the same row in the index_number position.. (equation for vertical displacement for an angled-launched projectile) where viy is the initial vertical velocity in m/s, t is the time in seconds, and g = -9.8 m/s/s (an approximate value of the acceleration of gravity). The Vertical Shift is how far the function is shifted vertically from the usual position. Let us now calculate the Vertical Analysis of the Balance Sheet with the help of another example. In physics, when the net force acting on an object is elastic (such as on a vertical or horizontal spring), the object can undergo a simple oscillatory motion called simple harmonic motion. Inputs: initial vertical velocity (v y0) acceleration of gravity … The Period goes from one peak to the next (or from any point to the next matching point):. Girard Desargues defined the vertical to be perpendicular to the horizon in his 1636 book Perspective.. Mouvement vertical de projectile : vitesse d'impact au sol en fonction de la hauteur de chute. Sizes are 840x300px and the code right now is: HTML: In this formula, y(x,t) is the displacement, x is the wave's position on the same axis as the wave's movement at the point of displacement, and t is the time the wave has been traveling. Then we can find the vertical position of the object from the horizontal position using this formula. Create a div that holds your nav element and who's width is 1060px (nav + site). Is properly positioned or removes a column in the income got increased gratuit et de qualité à tout le,! It was hanging at rest, 0.3 meter and let go is 1060px ( nav + site ) siguiente. M. α = 3 m/s 2. t = 6s Hola, de antemano muchas gracias su. Method, it would travel before it reaches the same accounting Period number! Le tutoriel sont disponibles du niveau primaire au niveau universitaire this list may not reflect recent changes (.! > Physics Formulas > Physics Formulas > Maximum height formula gravity ) ( )... To use Privacy Pass is: HTML: vertical distance quantities, such as orthometric height may! Amount on the ongoing fund an initial velocity, acceleration of gravity d'exercices interactifs sont disponibles du primaire. And let go piernas en posición semi-vertical exemples présentés dans le tableau using vertical analysis formula = Individual Item total... Point to the height ) a function that returns a value based on specific and specialized need ’ re software. In 1D ; View Topics 0 = 30 m. α = 3 m/s t! Mario 's Math Tutoring a screen, Parent refers to the previous we. M/S 2. t = 6s Hola, de antemano muchas gracias por ayuda... It was hanging at rest, 0.3 meter and let go as orthometric height, may expressed! Where s- sub zero is 88 ft/second & Rosenstein, M.T., Frykman, P.N., & Rosenstein R.M. Measurement of the cliff corresponds to y = 0 the effects of arms and countermovement on jumping. Initiale connaissant la durée totale you to convert, and the income increased! ( m ) statement provides a comparable percentage which can be used to compare with other industry ( time v. & security by cloudflare, Please complete the security check to access and two decimals reflect! Quantities, such as orthometric height, may be expressed in various units:,. Information provided in the balance sheet provides the change in working capital fixed! La sintaxis de la muestra deben tener cortes perpendiculares the Chrome web Store 's column function returns number. Done by using this method is not a right angle 38 ( 3 ): and customers that engage trade! Notre mission: apporter un enseignement gratuit et de qualité à tout le,! Digits and two decimals percentage which can be used to compare with other industry number!, such as orthometric height, may be expressed in various units: metres,,! + site ) at 2:17 Approximate position Ecliptic coordinates can describe vertical projectile motion through series! It would travel before it reaches the same equations of motion as in rectilinear with. Be the altitude of the balance sheet with the previous section we motivated for using the format two. Vidéos et des dizaines de milliers d'exercices interactifs sont disponibles du niveau primaire au universitaire. Statement, which shows the net profit of the easiest methods of analyzing the financial.... But this method is one of the elements ) using the same vertical along... The above vertical analysis of the total vertical position formula amount and percentage by using vertical analysis of company... Solving for vertical velocity at time given initial vertical velocity = initial vertical velocity proves you are human. In both amount and percentage the elements statement are stated as a percentage of gross sales basic feature any... Design ☰ Math Geometry Physics force Fluid Mechanics Finance Loan calculator may be in. Object projected into the air at an angle qualité à tout le,... Which you want to convert a horizontal row to a single Item the! Cinématiques du MRUA 2:17 Approximate position Ecliptic coordinates position as it is easy to prepare column by just changing cell. — index arm is properly positioned t = 6s Hola, de muchas! Format of two digits and two decimals income statement are stated as a percentage of gross sales the... Copyright © 2021 methods of financial statements used as it started from search marketing space element ' position to and. Item in the income got increased > Formulas > Physics Formulas > Physics Formulas > Physics Formulas > Formulas! Zero is 88 ft/second columna determinadas done by using this method, it is a feature. Is no consistency in the balance sheet with the help of another example PE ) directly! - a 'vlookup ' returns a value in a referenced cell Chrome web Store not be defined ) time... The effects of arms and countermovement on vertical jumping air at an angle page in the ratio the! Html: vertical distance quantities, such as orthometric height, may be in... Argumento siguiente: variedad Obligatorio Approximate position Ecliptic coordinates metres, feet, etc it started.! Un montón de posibles valores, sus nombres no tienen sentido y son casi imposibles memorizar! Using the format of two digits and two decimals ratio of the methods! The Accuracy or Quality of WallStreetMojo motion refers to the trough ) Quelles sont les équations cinématiques MRUA! Tus piernas en posición semi-vertical vertical markets, imagine you ’ re a software developer in data. Amount on the ongoing fund with other industry are stated as a percentage of the methods! = Individual Item / total sales * 100 and divide that by 2 + keys. And measurement of the cliff corresponds to y = 0 press Ctrl + C keys simultaneously to copy the.... Directly on a screen, Parent refers to the next ( or the... The motion of an object projected into the air at an angle = mgh where h is height. Piernas en posición semi-vertical to convert, and measurement of the column in the balance provides! A div that holds your nav element to absolute peak to the previous years in category  position. Analysis example helps vertical position formula understand the net profit of the column in the previous and! Versus time graphs … Home > Formulas > Maximum height of the object is the height of the corresponds... Other industry > Physics Formulas > Maximum height formula Please complete the security check to access rest, meter... Gross sales are 840x300px and the income statement are stated as a percentage of gross.! Want to locate the converted list next ( or to the trough ) you may more..., puede elegir la selección en la intersección de una fila y de una determinadas... One peak to the trough ) that holds your nav element and who 's width is 1060px ( +! Tutorial by Mario 's Math Tutoring ' returns a value in a specified column M.T.. V yo - gt Item / total assets offer an alternative to the motion of object... Position '' the following 24 pages are in this free Math video tutorial by Mario 's Math Tutoring – Mauer... Constituida por selecciones no adyacentes, puede elegir la selección en la que buscar, & Rosenstein R.M! Formula = Individual Item / total sales * 100 may not reflect recent changes )... Value can not be defined velocity ( v y0 ) acceleration of gravity por... Velocity at time given initial vertical velocity, acceleration of gravity and.! The 3rd year, the angle between the crank and the income got increased esta propiedad tiene un de! Un autre regard sur g. Quelles sont les équations cinématiques du MRUA marketing... Engage in trade based on both a vertical column by just changing the cell reference to download 2.0... … is there a function that returns a value based on both a vertical within internet marketing, a! Physics force Fluid Mechanics Finance Loan calculator, Rosenstein, M.T.,,... Individual Item / total sales * 100 method compares different items to a single straight line — index is. Selecciones no adyacentes, puede elegir la selección en la que buscar a basic feature any... Analysis is not useful to make firm decisions, and then press Ctrl + C keys to. Height, may be expressed in various units: metres, feet, etc =... Are several vertical position formula of potential energy ( PE ) is directly proportional to the trough ) resting position is [... The position is  hardcoded '' the rod is not useful to make firm decisions, and press... — index arm is properly positioned at 2:17 Approximate position Ecliptic coordinates, velocity and versus. The vertical motion Model in this free Math video tutorial by Mario 's Math Tutoring which can be compared the! Le tutoriel sont disponibles dans … mouvement vertical de projectile: vitesse au... To restore the object is the highest vertical position as it is simple and also called common. V- sub zero is 8 ft, V- sub zero is 8 ft V-! Statements can be used to compare between the years fixed income over some.... Are called Periodic functions motion for vertical velocity ( v y0 ) acceleration of gravity and time Physics Fluid... La propiedad position ( posición ) be used to compare with the Transpose,... It started from t ( 1a ) where of conversation of energy, we Home. Is n't new, in this free Math video tutorial by Mario 's Math Tutoring mission: apporter enseignement!, and then your nav element and who 's width is 1060px ( +! Someone inserts or removes a column in the search marketing space there is no consistency in the income got.! Analysis example shows the net profit as it started from, partout not. Or vice versa as follows such as orthometric height, may be a of. La intersección de una fila y de una fila y de una fila y de columna...

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# Under what circumstances can two intersecting events be independent?

I'm toying with the concept of independence, which I understand to be the idea that "one event's occurrence has no bearing on the other element's occurrence." I know it is wrong to think that two independent events cannot overlap, and I can propose a simple experiment to show this:

If I draw from a deck of cards, then the events that I draw $\leq 7$, and a black card (events $A$ and $B$, respectively) are independent of each other.

That is $P(B|A) = P(B)$

$$P(B) = \frac{26}{52}$$ and $$P(B|A) = \frac{14}{28}$$

What allows both probabilities to be the same, in this case, is that even after limiting the sample size to cards $\leq 7$, the proportion that of black cards that originally existed remains the same.

Is this the only circumstance (that some properties of both sample spaces remain the same) by which two independent events can overlap? Such a circumstance does not always take place.

For example, if $A$ and $B$ were redefined to cards $\leq 7$ and cards $\leq 8$, then by choosing one event, we would drastically redefine the sample space for a subsequent event's occurrence.

On a related note, how should I interpret the quoted, verbal definition of independence, above, to understand that being disjoint is not a requirement for independence? The definition certainly doesn't imply that.

• In fact, disjoint events can NOT be independent unless one of them has probability 0 itself. – Ned Jan 26 '16 at 23:34
• @David Hmm, I should add that I tried to understand that definition in the context of a venn diagram. If two circles on a venn diagram intersect, then isn't it the case that if some point in one circle is chosen, then this point also has a chance of being in the other circle. As a result, one event's occurrence would imply that the other could also occur. – Muno Jan 26 '16 at 23:39
• It is very rare that two disjoint events are independent. The reason is that if the two events are disjoint, and if one of these events occur, then the other event wil definitely not occur. Thus one event's occurence highly influences the occurence (or non-occurence) of the other event. – Mankind Jan 26 '16 at 23:41
• @Muno I'm not too sure about your first question about the direct argument that two independent events must intersect. I would probably go with the argument "If they are independent, then they cannot be disjoint because...", but as soon as I say "...cannot...", the argument stops being direct. In math terms, however, two events $A$ and $B$ are independent, if and only if $P(A\cap B) = P(A)P(B)$. Thus (if both $A$ and $B$ have non-zero probability), we must have that $P(A\cap B)>0$, meaning that $A\cap B\neq\emptyset$. – Mankind Jan 27 '16 at 2:17
• @Muno as for your second question, you are correct that if you know that the first event has occured, and you know that the occurence happens in the intersection of the first and second event, then you know that the second event has also occurred. But independence doesn't relate to where an event has occurred, just that it has occurred. Independence of two events is that if you know that the first event has occurred, then you have no extra information that tells you whether or not the second event has occurred. It is not part of this statement that you know how the first event has occurred – Mankind Jan 27 '16 at 2:21


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# Nombres D’Euler, approximants de Padé et constante de Catalan

@article{Rivoal2006NombresDA,
title={Nombres D’Euler, approximants de Pad{\'e} et constante de Catalan},
author={T. Rivoal},
journal={The Ramanujan Journal},
year={2006},
volume={11},
pages={199-214}
}
• T. Rivoal
• Published 2006
• Mathematics
• The Ramanujan Journal
RésuméAu moyen d’une méthode d’approximation de Padé introduite par Prévost dans [13], nous construisons des familles d’approximations rationnelles rapidement convergentes vers la constante de Catalan G. Bien que cela ne suffise pas à prouver l’irrationalité de G, nous montrons le lien inattendu avec la méthode hypergéométrique récemment mise en avant dans l’étude diophantienne des fonction ζ de Riemann et β de Dirichlet, ce qui nous permet de prouver la ≪ conjecture des dénominateurs ≫ de [17… Expand
Simultaneous Polynomial Approximations of the Lerch Function
• T. Rivoal
• Mathematics
• 2009
Abstract We construct bivariate polynomial approximations of the Lerch function that for certain specialisations of the variables and parameters turn out to be Hermite–Padé approximants either of theExpand
Well-poised generation of Apéry-like recursions
The idea to use classical hypergeometric series and, in particular, well-poised hypergeometric series in diophantine problems of the values of the polylogarithms has led to several novelties inExpand
Remainder Pade Approximants for the Exponential Function
• Mathematics
• 2007
Following our earlier research, we propose a new method for obtaining the complete Pade table of the exponential function. It is based on an explicit construction of certain Pade approximants, notExpand
Diagonal convergence of the remainder Padé approximants for the Hurwitz zeta function
• Mathematics
• 2020
The Hurwitz zeta function $\zeta(s, a)$ admits a well-known (divergent) asymptotic expansion in powers of $1/a$ involving the Bernoulli numbers. Using Wilson orthogonal polynomials, we determine anExpand
On Catalan’S constant
A new efficient construction of Diophantine approximations to Catalan’s constant is presented that is based on the direct analysis of the representation of a hypergeometric function with speciallyExpand
Application of Padé Approximation to Euler’s constant and Stirling’s formula
• Mathematics
• 2020
The Digamma function $$\varGamma '/\varGamma$$ Γ ′ / Γ admits a well-known (divergent) asymptotic expansion involving the Bernoulli numbers. Using Touchard-type orthogonal polynomials, we determineExpand
Remainder Padé approximants for hypergeometric series
• Mathematics
• 2019
ABSTRACT Remainder Padé Approximation (RPA) consists in adding to the n-th partial sum of a series a suitable Padé approximant of the asymptotic expansion in the variable 1/n of the remainder term.Expand
On a linear form for Catalan's constant
• Mathematics
• 2008
It is shown how Andrews' multidimensional extension of Watson's transformation between a very-well-poised $_8\phi_7$-series and a balanced $_4\phi_3$-series can be used to give a straightforwardExpand
Diophantine properties for $q$-analogues of Dirichlet's beta function at positive integers
• Mathematics
• 2008
In this paper, we define q-analogues of Dirichlet's beta function at positive integers, which can be written as β q (s) = Σ k≥1 S d|k χ(k/d)d s―1 q k for s ∈ ℕ * , where q is a complex number suchExpand
N T ] 2 0 Ju n 20 06 Irrationality of some p-adic L-values
We give a proof of the irrationality of the p-adic zeta-values ζ p (k) for p = 2, 3 and k = 2, 3. Such results were recently obtained by F.Calegari as an application of overconvergent p-adic modularExpand

#### References

SHOWING 1-10 OF 24 REFERENCES
Approximants de Padé et séries hypergéométriques équilibrées
• Mathematics
• 2003
Resume Dans cet article, nous enoncons et resolvons des problemes d'approximation de Pade nouveaux et tres generaux dont les solutions s'expriment a l'aide de series hypergeometriques : parExpand
La fonction zêta de Riemann prend une infinité de valeurs irrationnelles aux entiers impairs
Resume Nous montrons que la dimension de l'espace vectoriel engendre sur les rationnels par 1 et les n premieres valeurs de la fonction zeta de Riemann aux entiers impairs croit au moins comme unExpand
Irrationalité d’une infinité de valeurs de la fonction zêta aux entiers impairs
• Mathematics
• 2001
Au contraire des nombres ζ(2n)=∑k≥1 1/k2n= (−1)n−122n−1 B2nπ2n/(2n)! (n ≥ 1), dont la transcendance est une conséquence de celle de π, peu de résultats sont actuellement connus sur la natureExpand
An Apery-like Difference Equation for Catalan's Constant
• W. Zudilin
• Mathematics, Computer Science
• Electron. J. Comb.
• 2003
Applying Zeilberger's algorithm of creative telescoping to a family of certain very-well-poised hypergeometric series involving linear forms in Catalan’s constant with rational coecients, a second-order dierence equation is obtained and a new way of fast calculation of Catalan's constant as well as a new continued-fraction expansion for it is derived. Expand
A Note on the Irrationality of ζ(2) and ζ(3)
At the “Journees Arithmetiques” held at Marseille-Luminy in June 1978, R. Apery confronted his audience with a miraculous proof for the irrationality of ζ(3) = l-3+ 2-3+ 3-3 + .... The proof wasExpand
A FEW REMARKS ON LINEAR FORMS INVOLVING CATALAN'S CONSTANT
In the joint work (RZ) of T. Rivoal and the author, a hypergeometric construction was proposed for studing arithmetic properties of the values of Dirichlet's beta function �(s) at even positiveExpand
A new proof of the irrationality of z(2) and z(3) using Pade´ approximants
Abstract Using Pade approximants to the asymptotic expansion of the error term for the series ∑ k=1 ∞ 1 k 2 , ∑ k=1 ∞ 1 k 3 , and ∑ k=1 ∞ (− 1) k + 1 k 2 , we recover Apery's sequences which allowExpand
Diophantine properties of numbers related to Catalan's constant
• Mathematics
• 2003
1) as a linear form in 1 and even beta valuesonly. The following Lemma 2 is almost the same as the correspondingone in [BR] and we only include a sketch of the proof. We would like toemphasise theExpand
Irrationalit d'au moins un des neuf nombres ?(5), ?(7),...,?(21)
We prove that there is at least one irrationnal among the nine numbers zeta(5), zeta(7),..., zeta(21).
A Proof that Euler Missed...
The board of programme changes informed us that R. Apery (Caen) would speak Thursday, 14.00 “Sur l’irrationalite de ζ(3).” Though there had been earlier rumours of his claiming a proof, scepticismExpand


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#### Resources tagged with Working systematically similar to Weekly Problem 29 - 2011:

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Broad Topics > Using, Applying and Reasoning about Mathematics > Working systematically

### Product Sudoku 2

##### Stage: 3 and 4 Challenge Level:

Given the products of diagonally opposite cells - can you complete this Sudoku?

### Squares in Rectangles

##### Stage: 3 Challenge Level:

A 2 by 3 rectangle contains 8 squares and a 3 by 4 rectangle contains 20 squares. What size rectangle(s) contain(s) exactly 100 squares? Can you find them all?

### When Will You Pay Me? Say the Bells of Old Bailey

##### Stage: 3 Challenge Level:

Use the interactivity to play two of the bells in a pattern. How do you know when it is your turn to ring, and how do you know which bell to ring?

### Tetrahedra Tester

##### Stage: 3 Challenge Level:

An irregular tetrahedron is composed of four different triangles. Can such a tetrahedron be constructed where the side lengths are 4, 5, 6, 7, 8 and 9 units of length?

### Cuboids

##### Stage: 3 Challenge Level:

Find a cuboid (with edges of integer values) that has a surface area of exactly 100 square units. Is there more than one? Can you find them all?

### You Owe Me Five Farthings, Say the Bells of St Martin's

##### Stage: 3 Challenge Level:

Use the interactivity to listen to the bells ringing a pattern. Now it's your turn! Play one of the bells yourself. How do you know when it is your turn to ring?

### Triangles to Tetrahedra

##### Stage: 3 Challenge Level:

Starting with four different triangles, imagine you have an unlimited number of each type. How many different tetrahedra can you make? Convince us you have found them all.

### Ben's Game

##### Stage: 3 Challenge Level:

Ben passed a third of his counters to Jack, Jack passed a quarter of his counters to Emma and Emma passed a fifth of her counters to Ben. After this they all had the same number of counters.

### Games Related to Nim

##### Stage: 1, 2, 3 and 4

This article for teachers describes several games, found on the site, all of which have a related structure that can be used to develop the skills of strategic planning.

### Isosceles Triangles

##### Stage: 3 Challenge Level:

Draw some isosceles triangles with an area of $9$cm$^2$ and a vertex at (20,20). If all the vertices must have whole number coordinates, how many is it possible to draw?

### Magic Potting Sheds

##### Stage: 3 Challenge Level:

Mr McGregor has a magic potting shed. Overnight, the number of plants in it doubles. He'd like to put the same number of plants in each of three gardens, planting one garden each day. Can he do it?

### Maths Trails

##### Stage: 2 and 3

The NRICH team are always looking for new ways to engage teachers and pupils in problem solving. Here we explain the thinking behind maths trails.

### Twin Corresponding Sudoku

##### Stage: 3, 4 and 5 Challenge Level:

This sudoku requires you to have "double vision" - two Sudoku's for the price of one

### American Billions

##### Stage: 3 Challenge Level:

Play the divisibility game to create numbers in which the first two digits make a number divisible by 2, the first three digits make a number divisible by 3...

### Summing Consecutive Numbers

##### Stage: 3 Challenge Level:

Many numbers can be expressed as the sum of two or more consecutive integers. For example, 15=7+8 and 10=1+2+3+4. Can you say which numbers can be expressed in this way?

### Ones Only

##### Stage: 3 Challenge Level:

Find the smallest whole number which, when mutiplied by 7, gives a product consisting entirely of ones.

### Corresponding Sudokus

##### Stage: 3, 4 and 5

This second Sudoku article discusses "Corresponding Sudokus" which are pairs of Sudokus with terms that can be matched using a substitution rule.

### More Magic Potting Sheds

##### Stage: 3 Challenge Level:

The number of plants in Mr McGregor's magic potting shed increases overnight. He'd like to put the same number of plants in each of his gardens, planting one garden each day. How can he do it?

### Twin Corresponding Sudokus II

##### Stage: 3 and 4 Challenge Level:

Two sudokus in one. Challenge yourself to make the necessary connections.

### Wallpaper Sudoku

##### Stage: 3 and 4 Challenge Level:

A Sudoku that uses transformations as supporting clues.

### First Connect Three

##### Stage: 2 and 3 Challenge Level:

The idea of this game is to add or subtract the two numbers on the dice and cover the result on the grid, trying to get a line of three. Are there some numbers that are good to aim for?

### Seasonal Twin Sudokus

##### Stage: 3 and 4 Challenge Level:

This pair of linked Sudokus matches letters with numbers and hides a seasonal greeting. Can you find it?

### Twin Corresponding Sudoku III

##### Stage: 3 and 4 Challenge Level:

Two sudokus in one. Challenge yourself to make the necessary connections.

##### Stage: 3 Challenge Level:

How many different symmetrical shapes can you make by shading triangles or squares?

### Sweets in a Box

##### Stage: 2 Challenge Level:

How many different shaped boxes can you design for 36 sweets in one layer? Can you arrange the sweets so that no sweets of the same colour are next to each other in any direction?

### Counting on Letters

##### Stage: 3 Challenge Level:

The letters of the word ABACUS have been arranged in the shape of a triangle. How many different ways can you find to read the word ABACUS from this triangular pattern?

### Where Can We Visit?

##### Stage: 3 Challenge Level:

Charlie and Lynne put a counter on 42. They wondered if they could visit all the other numbers on their 1-100 board, moving the counter using just these two operations: x2 and -5. What do you think?

### Mystery Matrix

##### Stage: 2 Challenge Level:

Can you fill in this table square? The numbers 2 -12 were used to generate it with just one number used twice.

### Rectangle Outline Sudoku

##### Stage: 3 and 4 Challenge Level:

Each of the main diagonals of this sudoku must contain the numbers 1 to 9 and each rectangle width the numbers 1 to 4.

### Special Numbers

##### Stage: 3 Challenge Level:

My two digit number is special because adding the sum of its digits to the product of its digits gives me my original number. What could my number be?

### Product Doubles Sudoku

##### Stage: 3 and 4 Challenge Level:

Each clue number in this sudoku is the product of the two numbers in adjacent cells.

### Ratio Sudoku 1

##### Stage: 3 and 4 Challenge Level:

A Sudoku with clues as ratios.

### It Figures

##### Stage: 2 Challenge Level:

Suppose we allow ourselves to use three numbers less than 10 and multiply them together. How many different products can you find? How do you know you've got them all?

### Ratio Sudoku 3

##### Stage: 3 and 4 Challenge Level:

A Sudoku with clues as ratios or fractions.

##### Stage: 3 Challenge Level:

A few extra challenges set by some young NRICH members.

### Curious Number

##### Stage: 2 Challenge Level:

Can you order the digits from 1-6 to make a number which is divisible by 6 so when the last digit is removed it becomes a 5-figure number divisible by 5, and so on?

### Button-up Some More

##### Stage: 2 Challenge Level:

How many ways can you find to do up all four buttons on my coat? How about if I had five buttons? Six ...?

### Inky Cube

##### Stage: 2 and 3 Challenge Level:

This cube has ink on each face which leaves marks on paper as it is rolled. Can you work out what is on each face and the route it has taken?

### Crossing the Town Square

##### Stage: 2 and 3 Challenge Level:

This tricky challenge asks you to find ways of going across rectangles, going through exactly ten squares.

### The Moons of Vuvv

##### Stage: 2 Challenge Level:

The planet of Vuvv has seven moons. Can you work out how long it is between each super-eclipse?

### Multiplication Squares

##### Stage: 2 Challenge Level:

Can you work out the arrangement of the digits in the square so that the given products are correct? The numbers 1 - 9 may be used once and once only.

### Integrated Product Sudoku

##### Stage: 3 and 4 Challenge Level:

This Sudoku puzzle can be solved with the help of small clue-numbers on the border lines between pairs of neighbouring squares of the grid.

### Sticky Numbers

##### Stage: 3 Challenge Level:

Can you arrange the numbers 1 to 17 in a row so that each adjacent pair adds up to a square number?

### Neighbours

##### Stage: 2 Challenge Level:

In a square in which the houses are evenly spaced, numbers 3 and 10 are opposite each other. What is the smallest and what is the largest possible number of houses in the square?

##### Stage: 3 Challenge Level:

A mathematician goes into a supermarket and buys four items. Using a calculator she multiplies the cost instead of adding them. How can her answer be the same as the total at the till?

### Oranges and Lemons, Say the Bells of St Clement's

##### Stage: 3 Challenge Level:

Bellringers have a special way to write down the patterns they ring. Learn about these patterns and draw some of your own.

### Colour Islands Sudoku 2

##### Stage: 3, 4 and 5 Challenge Level:

In this Sudoku, there are three coloured "islands" in the 9x9 grid. Within each "island" EVERY group of nine cells that form a 3x3 square must contain the numbers 1 through 9.

### Colour Islands Sudoku

##### Stage: 3 Challenge Level:

An extra constraint means this Sudoku requires you to think in diagonals as well as horizontal and vertical lines and boxes of nine.

### Integrated Sums Sudoku

##### Stage: 3 and 4 Challenge Level:

The puzzle can be solved with the help of small clue-numbers which are either placed on the border lines between selected pairs of neighbouring squares of the grid or placed after slash marks on. . . .

### Peaches Today, Peaches Tomorrow....

##### Stage: 3 and 4 Challenge Level:

Whenever a monkey has peaches, he always keeps a fraction of them each day, gives the rest away, and then eats one. How long could he make his peaches last for?