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Profit maximization is the process of identifying the most efficient manner of obtaining the highest rate of return from its production model. Profit Maximization Model. In this approach actions that increase the profits â¦ Maximize definition is - to increase to a maximum. How to use maximize in a sentence. It simply means maximization of shareholderâs wealth. The ultimate objective of any business is to earn a huge amount of return in terms of profit. Profit Margin Cons Though profit maximization is an essential strategy for businesses, there are still disadvantages to using this model. To be even more meticulous, a shareholder holds share in the company/business and his wealth will improve if the share price in the market increases which in turn is a function of net worth. While earning a profit is the goal of every business, profit maximization in financial management can put too much emphasis on profits and not enough emphasis on other aspects of the business such as customer retention, social and economic well-being, and other goals and aspects of the company. Maximize definition, to increase to the greatest possible amount or degree: to look for ways of maximizing profit. maximización de beneficios. In economics, it is always assumed that a firmâs rationality is the maximization of profit. Profit maximization refers to the sales level where profits are highest. Profit is a financial benefit that is realized when the amount of revenue gained from a business activity exceeds the expenses, costs, and taxes needed to sustain the activity. Because shareholders own the firm, they are entitled to the profits of the firm. Profit Maximization avoids time value of money, but Wealth Maximization recognises it. Profit maximization is a short term objective of the firm while the long-term objective is Wealth Maximization. 2. Wealth maximization is the concept of increasing the value of a business in order to increase the value of the shares held by its stockholders. Profit maximization as the name suggests refers to that strategy in which company strives to achieve maximum possible profit by selling the goods or service at highest possible price at the same time ensuring that cost of production is kept low whereas wealth maximization refers to that strategy in which company strive to increase the value of the business in the long term even if it involves sacrificing profits in the short term. Features of Profit Maximization â Firms choose investment proposals which suits profit maximization criteria and reject proposals which bring less profit. Profit maximization refers to plans and activities involved in the company's effort to boost net profit to the highest possible degree given the company's current resources. $$\text{0}=\text{MR}\ - \text{MC}$$ $$\text{MR}=\text{MC}$$ The beauty of MR = MC as the profit maximization point is that it applies to all firms, both in â¦ Decreasing Profit by Increasing Quantity. A wealth of a shareholder maximizes when the net worth of a company maximizes. Neoclassical economics, currently the mainstream approach to microeconomics, usually models the firm as maximizing profit.. It is also possible to focus on more long-term measures, such as the amount of equity versus debt. Profit maximization refers to the maximization of dollar income of the firm. It is a combination of two words viz. Type: noun; Copy to clipboard; Details / edit; Termium. This means selling a quantity of a good or service, or fixing a price, where total revenue (TR) is at its greatest above total cost (TC). Profit maximization is one of the many goals of financial management. MC = MR. Profit Maximization Example Explanation: Profit maximization is assumed to be the business objective of most firms in Economics unless specified otherwise. maximization. What Is Profit Maximization? In economics, profit maximization is the short run or long run process by which a firm may determine the price, input, and output levels that lead to the highest profit. Journal of Economic Issues: Vol. Profit earning capacity is kind of a parameter for measuring the â¦ Any profit â¦ Stepping stone to success: Short-term success doesnât mean anything if you canât sustain it. It is a very simple and unambiguous model. Profit maximisation is the process that companies undergo in order to determine the best output and price levels in order to achieve its goals. Under profit maximization objective, business firms attempt to adopt those investment projects, which yields â¦ It is possible for a company to focus on more short-term measures of success such as quarterly profits. wealth and maximization. In this diagram, profit is maximised at â¦ Profit maximization is a necessity to both the survival and growth of your business. Profit maximisation is one of the fundamental assumptions of economic theory. For example, the term profit may mean long-term profit or short-term profit, profit after tax or profit before tax, gross profit or net profit, earning per share, return on equity etc. The concept requires a company's management team to continually search for the highest possible returns on funds invested â¦ Profit Maximization consists of thefollowing features: 1. Profit Maximization ignores risk and uncertainty. Shareholders do. There are several different approaches to this pursuit that may be used by any corporation or business. The difference between value maximization and profit maximization is mainly a concern of publicly traded companies. Marginal Revenue is also the slope of Total Revenue. 6, No. Profit Maximization model helps to predict the price-output behavior of a firm under changing market conditions like tax rates, wages and salaries, bonus, the degree of availability of resources, technology, fashions, tastes and preferences of consumers etc. These are the individuals, businesses, and institutions that have an ownership interest in a company after purchasing shares of that company's stock. It helps in achieving the objects to maximize the business operation for profit maximization. If the company were to keep increasing output past the â¦ The profit maximization rule formula is MC = MR Marginal Costis the increase in cost by producing one more unit of the good. Profit maximisation â definition Profit maximisation is assumed to be the dominant goal of a typical firm. We need a more balanced form of â¦ This is because wealth maximization is also knoâ¦ There are several approaches to this problem. Looking from a producerâs perspective, profit is their favorite word in the book of economics. profit maximization . 61-66. the process of making something as great in amount, size, or importance as possible: Short term profit maximization doesn't necessarily increase shareholder value. Even if your business is a one-person shop, you are the shareholder because of your invested interest in your company. So if profit maximization is taken as a goal of the firm, there will be confusing in decision-making. Profit Maximization is also known as cash per share maximization. Profit maximization is one of the most important assumptions of economic theory. Who owns a corporation? So, it becomes the most crucial goal of the company to survive and grow in the current cut-throat competitive landscape of â¦ In economics, profit maximization is the process by which a firm determines the price and output level that returns the greatest profit. It means, rational producer or entrepreneur always attempts for profit maximization. Profit = Total Revenue â Total Costs Therefore, profit maximization occurs at the most significant gap or the biggest difference between the total revenue and the total cost. Thus it is easy to realize that a producer would be in the state of equilibrium if he is earning maximum profit, i.e has profit maximisation. Profit maximization: Profit maximization is considered as the goal of financial management. Profit Maximization is the ability of the company to operate efficiently to produce maximum output with limited input or to produce the same output using much lesser input. What Is Profit Maximization? Profit Maximization Definition: Profit Maximization is defined as producing at the quantity where a firmâs marginal cost equals marginal revenues i.e. 3. Equilibrium represents a state of no change. But âTR/âq is the definition of marginal revenue (MR) and âTC/âq is the definition of marginal cost. You might have seen the profit maximization formula presented in economics textbooks as: Marginal Cost = Marginal Revenue In simpler terms, profit maximization occurs when the profits are highest at a certain number of sales. But profit maximization does teach business owners about the importance of being profitable from Day 1. Competitive equilibrium is a condition in which profit-maximizing producers and utility-maximizing consumers in competitive markets with freely determined prices arrive at â¦ In the jargon of economists, profit maximization occurs when marginal cost is equal to marginal revenue. It is the traditional approach and the primary objective of financial management. Unlike Wealth Maximization, which considers both. There are several perspectives one can take on this problem. In economics, profit maximization is the short run or long run process by which a firm determines the price and output level that returns the greatest profit. Under such approach maximization of profit is the sole objective of a business and the behavior of a firm is analyzed in terms of its profit maximization ability. translation and definition "profit maximization", English-Spanish Dictionary online. There are several approaches to this problem. Profit Maximization is necessary for the survival and growth of the enterprise. Currently, government is left to address the problems created by profit-maximizing individuals and businesses. It will be achieved when a firm reaches the stage of equilibrium. [ËmæksÉªmaÉªËzeÉªÊÉn] N [ of profits, assets, potential] â maximización f. Collins Spanish Dictionary - Complete and Unabridged 8th Edition 2005 © William Collins Sons & Co. Ltd. 1971, 1988 © HarperCollins Publishers 1992, 1993, 1996, 1997, 2000, 2003, 2005. maximization. 2-3, pp. Thus, this objective of financial management considers all the possible ways to increase the profitability of the business concern. Marginal Revenue is the change in total revenueas a result of changing the rate of sales by one unit. See more. (1972).
2020 profit maximization meaning


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Sample-Efficient Imitation Learning via Generative Adversarial Nets

Generative Adversarial Imitation Learning (GAIL) [1] is a recent successful imitation learning architecture that exploits the adversarial training procedure introduced in Generative Adversarial Networks (GAN) [2]. Albeit successful at generating behaviours similar to those demonstrated to the agent, GAIL suffers from a high sample complexity in the number of interactions it has to carry out in the environment in order to achieve satisfactory performance. We dramatically shrink the amount of interactions with the environment necessary to learn well-behaved imitation policies, by up to several orders of magnitude. Our framework, operating in the model-free regime, exhibits a significant increase in sample-efficiency over previous methods by simultaneously a) learning a self-tuned adversarially-trained surrogate reward and b) leveraging an off-policy actor-critic architecture. We show that our approach is simple to implement and that the learned agents remain remarkably stable, as shown in our experiments that span a variety of continuous control tasks.

Paper

Lionel Blondé, Alexandros Kalousis.

We address the problem of an agent learning to act in an environment in order to reproduce the behaviour of an expert demonstrator. No direct supervision is provided to the agent (she is never directly told what the optimal action is) nor does she receives a reinforcement signal from the environment upon interaction. Instead, the agent is provided with a pool of trajectories.

The approach in this paper, named Sample-efficient Adversarial Mimic (SAM), adopts an off-policy TD learning paradigm. By storing past experiences and replaying them in an uncorrelated fashion, SAM displays significant gains in sample-efficiency. To solve the differentiability bottleneck of [1] caused by the stochasticity of its generator, we operate over deterministic policies. By relying on an off-policy actor-critic architecture and wielding deterministic policies, SAM builds on the Deep Deterministic Policy Gradients (DDPG) algorithm [4], in the context of Imitation Learning.

Inter-module relationships in different neural architectures (the scope of this figure was inspired from [3]). Modules with distinct loss functions are depicted with empty circles, while filled circles designate environmental entities. Solid and dotted arrows respectively represent (forward) flow of information and (backward) flow of gradient. a) Generative Adversarial Imitation Learning [1] b) Actor-Critic architecture [5] c) SAM (this work).

SAM is composed of three interconnected learning modules: a reward module (parameter $$\phi$$), a policy module (parameter $$\theta$$), a critic module (parameter $$\psi$$). The reward and policy modules are both involved in a GAN's adversarial training procedure, while the policy and critic modules are trained as an actor-critic architecture. As reminded recently in [3], GANs and actor-critic architectures can be both framed as bilevel optimization problems, each involving two competing components, which we just listed out for both architectures. Interestingly, the policy module plays a role in both problems, tying the two bilevel optimization problems together. In one problem, the policy module is trained against the reward module, while in the other, the policy module is trained against the critic module. The reward and critic modules can therefore be seen as serving analogous roles in their respective bilevel optimization problems: forging and maintaining a signal which enables the reward-seeking policy to adopt the desired behaviour. How each of these component is optimised is described in the paper.

Results

Performance comparison between SAM and GAIL in terms of episodic return. The horizontal axis depicts, in logarithmic scale, the number of interactions with the environment.

References

[1] Ho, Jonathan, and Stefano Ermon. 2016. “Generative Adversarial Imitation Learning.” In Neural Information Processing Systems (NIPS). link.

[2] Goodfellow, Ian, Jean Pouget-Abadie, Mehdi Mirza, Bing Xu, David Warde-Farley, Sherjil Ozair, Aaron Courville, and Yoshua Bengio. 2014. “Generative Adversarial Nets.” In Neural Information Processing Systems (NIPS).

[3] Pfau, David, and Oriol Vinyals. 2016. “Connecting Generative Adversarial Networks and Actor-Critic Methods.” arXiv [cs.LG]. link.

[4] Lillicrap, Timothy P., Jonathan J. Hunt, Alexander Pritzel, Nicolas Heess, Tom Erez, Yuval Tassa, David Silver, and Daan Wierstra. 2016. “Continuous Control with Deep Reinforcement Learning.” In International Conference on Learning Representations (ICLR). link.

[5] Sutton, Richard S., David A. McAllester, Satinder P. Singh, and Yishay Mansour. 1999. “Policy Gradient Methods for Reinforcement Learning with Function Approximation.” In Neural Information Processing Systems (NIPS).

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# Volume 17, Issue 1

### 1. Axiomatizing Maximal Progress and Discrete Time

Milner's complete proof system for observational congruence is crucially based on the possibility to equate $\tau$ divergent expressions to non-divergent ones by means of the axiom $recX. (\tau.X + E) = recX. \tau. E$. In the presence of a notion of priority, where, e.g., actions of type $\delta$ have a lower priority than silent $\tau$ actions, this axiom is no longer sound. Such a form of priority is, however, common in timed process algebra, where, due to the interpretation of $\delta$ as a time delay, it naturally arises from the maximal progress assumption. We here present our solution, based on introducing an auxiliary operator $pri(E)$ defining a "priority scope", to the long time open problem of axiomatizing priority using standard observational congruence: we provide a complete axiomatization for a basic process algebra with priority and (unguarded) recursion. We also show that, when the setting is extended by considering static operators of a discrete time calculus, an axiomatization that is complete over (a characterization of) finite-state terms can be developed by re-using techniques devised in the context of a cooperation with Prof. Jos Baeten.

### 2. Tameness in least fixed-point logic and McColm's conjecture

We investigate four model-theoretic tameness properties in the context of least fixed-point logic over a family of finite structures. We find that each of these properties depends only on the elementary (i.e., first-order) limit theory, and we completely determine the valid entailments among them. In contrast to the context of first-order logic on arbitrary structures, the order property and independence property are equivalent in this setting. McColm conjectured that least fixed-point definability collapses to first-order definability exactly when proficiency fails. McColm's conjecture is known to be false in general. However, we show that McColm's conjecture is true for any family of finite structures whose limit theory is model-theoretically tame.

### 3. 2-adjoint equivalences in homotopy type theory

We introduce the notion of (half) 2-adjoint equivalences in Homotopy Type Theory and prove their expected properties. We formalized these results in the Lean Theorem Prover.

### 4. On the Strong Equivalences for LPMLN Programs

LPMLN is a powerful knowledge representation and reasoning tool that combines the non-monotonic reasoning ability of Answer Set Programming (ASP) and the probabilistic reasoning ability of Markov Logic Networks (MLN). In this paper, we study the strong equivalence for LPMLN programs, which is an important tool for program rewriting and theoretical investigations in the field of logic programming. First of all, we present the notion of p-strong equivalence for LPMLN and present a model-theoretical characterization for the notion. And we investigate the relationships among the p-strong equivalence and other existing notions of strong equivalences for LPMLN. Then, we investigate several properties of the p-strong equivalence from the following four aspects. Firstly, we investigate two relaxed notions of the p-strong equivalence according to practical scenarios of program rewriting, and present corresponding characterizations for the notions. Secondly, we analyze the computational complexities of deciding strong equivalences for LPMLN programs. Thirdly, we investigate the relationships among the strong equivalences of LPMLN and two extensions of ASP: ASP with weak constraints and ordered disjunctions. Finally, we investigate LPMLN program simplification via the p-strong equivalence and present some syntactic conditions that decide the p-strong equivalence between a single LPMLN rule and the empty program. The contributions of the paper are as follows. Firstly, all of the […]

### 5. Definable decompositions for graphs of bounded linear cliquewidth

We prove that for every positive integer k, there exists an MSO_1-transduction that given a graph of linear cliquewidth at most k outputs, nondeterministically, some cliquewidth decomposition of the graph of width bounded by a function of k. A direct corollary of this result is the equivalence of the notions of CMSO_1-definability and recognizability on graphs of bounded linear cliquewidth.

### 6. Modal Logics for Nominal Transition Systems

We define a general notion of transition system where states and action labels can be from arbitrary nominal sets, actions may bind names, and state predicates from an arbitrary logic define properties of states. A Hennessy-Milner logic for these systems is introduced, and proved adequate and expressively complete for bisimulation equivalence. A main technical novelty is the use of finitely supported infinite conjunctions. We show how to treat different bisimulation variants such as early, late, open and weak in a systematic way, explore the folklore theorem that state predicates can be replaced by actions, and make substantial comparisons with related work. The main definitions and theorems have been formalised in Nominal Isabelle.

### 7. Solvability = Typability + Inhabitation

We extend the classical notion of solvability to a lambda-calculus equipped with pattern matching. We prove that solvability can be characterized by means of typability and inhabitation in an intersection type system P based on non-idempotent types. We show first that the system P characterizes the set of terms having canonical form, i.e. that a term is typable if and only if it reduces to a canonical form. But the set of solvable terms is properly contained in the set of canonical forms. Thus, typability alone is not sufficient to characterize solvability, in contrast to the case for the lambda-calculus. We then prove that typability, together with inhabitation, provides a full characterization of solvability, in the sense that a term is solvable if and only if it is typable and the types of all its arguments are inhabited. We complete the picture by providing an algorithm for the inhabitation problem of P.

### 8. Correct and Efficient Antichain Algorithms for Refinement Checking

The notion of refinement plays an important role in software engineering. It is the basis of a stepwise development methodology in which the correctness of a system can be established by proving, or computing, that a system refines its specification. Wang et al. describe algorithms based on antichains for efficiently deciding trace refinement, stable failures refinement and failures-divergences refinement. We identify several issues pertaining to the soundness and performance in these algorithms and propose new, correct, antichain-based algorithms. Using a number of experiments we show that our algorithms outperform the original ones in terms of running time and memory usage. Furthermore, we show that additional run time improvements can be obtained by applying divergence-preserving branching bisimulation minimisation.

### 9. Computing Probabilistic Bisimilarity Distances for Probabilistic Automata

The probabilistic bisimilarity distance of Deng et al. has been proposed as a robust quantitative generalization of Segala and Lynch's probabilistic bisimilarity for probabilistic automata. In this paper, we present a characterization of the bisimilarity distance as the solution of a simple stochastic game. The characterization gives us an algorithm to compute the distances by applying Condon's simple policy iteration on these games. The correctness of Condon's approach, however, relies on the assumption that the games are stopping. Our games may be non-stopping in general, yet we are able to prove termination for this extended class of games. Already other algorithms have been proposed in the literature to compute these distances, with complexity in $\textbf{UP} \cap \textbf{coUP}$ and \textbf{PPAD}. Despite the theoretical relevance, these algorithms are inefficient in practice. To the best of our knowledge, our algorithm is the first practical solution. The characterization of the probabilistic bisimilarity distance mentioned above crucially uses a dual presentation of the Hausdorff distance due to Mémoli. As an additional contribution, in this paper we show that Mémoli's result can be used also to prove that the bisimilarity distance bounds the difference in the maximal (or minimal) probability of two states to satisfying arbitrary $\omega$-regular properties, expressed, eg., as LTL formulas.

### 10. Determinacy in Discrete-Bidding Infinite-Duration Games

In two-player games on graphs, the players move a token through a graph to produce an infinite path, which determines the winner of the game. Such games are central in formal methods since they model the interaction between a non-terminating system and its environment. In bidding games the players bid for the right to move the token: in each round, the players simultaneously submit bids, and the higher bidder moves the token and pays the other player. Bidding games are known to have a clean and elegant mathematical structure that relies on the ability of the players to submit arbitrarily small bids. Many applications, however, require a fixed granularity for the bids, which can represent, for example, the monetary value expressed in cents. We study, for the first time, the combination of discrete-bidding and infinite-duration games. Our most important result proves that these games form a large determined subclass of concurrent games, where determinacy is the strong property that there always exists exactly one player who can guarantee winning the game. In particular, we show that, in contrast to non-discrete bidding games, the mechanism with which tied bids are resolved plays an important role in discrete-bidding games. We study several natural tie-breaking mechanisms and show that, while some do not admit determinacy, most natural mechanisms imply determinacy for every pair of initial budgets.

### 11. Axiomatization of betweenness in order-theoretic trees

The ternary betweenness relation of a tree, B(x,y,z) expresses that y is on the unique path between x and z. This notion can be extended to order-theoretic trees defined as partial orders such that the set of nodes larger than any node is linearly ordered. In such generalized trees, the unique "path" between two nodes can have infinitely many nodes. We generalize some results obtained in a previous article for the betweenness of join-trees. Join-trees are order-theoretic trees such that any two nodes have a least upper-bound. The motivation was to define conveniently the rank-width of a countable graph. We called quasi-tree the structure based on the betweenness relation of a join-tree. We proved that quasi-trees are axiomatized by a first-order sentence. Here, we obtain a monadic second-order axiomatization of betweenness in order-theoretic trees. We also define and compare several induced betweenness relations, i.e., restrictions to sets of nodes of the betweenness relations in generalized trees of different kinds. We prove that induced betweenness in quasi-trees is characterized by a first-order sentence. The proof uses order-theoretic trees.

### 12. Robustness Against Transactional Causal Consistency

Distributed storage systems and databases are widely used by various types of applications. Transactional access to these storage systems is an important abstraction allowing application programmers to consider blocks of actions (i.e., transactions) as executing atomically. For performance reasons, the consistency models implemented by modern databases are weaker than the standard serializability model, which corresponds to the atomicity abstraction of transactions executing over a sequentially consistent memory. Causal consistency for instance is one such model that is widely used in practice. In this paper, we investigate application-specific relationships between several variations of causal consistency and we address the issue of verifying automatically if a given transactional program is robust against causal consistency, i.e., all its behaviors when executed over an arbitrary causally consistent database are serializable. We show that programs without write-write races have the same set of behaviors under all these variations, and we show that checking robustness is polynomial time reducible to a state reachability problem in transactional programs over a sequentially consistent shared memory. A surprising corollary of the latter result is that causal consistency variations which admit incomparable sets of behaviors admit comparable sets of robust programs. This reduction also opens the door to leveraging existing methods and tools for the verification of concurrent […]

### 13. Convolution Algebras: Relational Convolution, Generalised Modalities and Incidence Algebras

Convolution is a ubiquitous operation in mathematics and computing. The Kripke semantics for substructural and interval logics motivates its study for quantale-valued functions relative to ternary relations. The resulting notion of relational convolution leads to generalised binary and unary modal operators for qualitative and quantitative models, and to more conventional variants, when ternary relations arise from identities over partial semigroups. Convolution-based semantics for fragments of categorial, linear and incidence (segment or interval) logics are provided as qualitative applications. Quantitative examples include algebras of durations and mean values in the duration calculus.

### 14. The $\pi$-Calculus is Behaviourally Complete and Orbit-Finitely Executable

Reactive Turing machines extend classical Turing machines with a facility to model observable interactive behaviour. We call a behaviour (finitely) executable if, and only if, it is equivalent to the behaviour of a (finite) reactive Turing machine. In this paper, we study the relationship between executable behaviour and behaviour that can be specified in the $\pi$-calculus. We establish that every finitely executable behaviour can be specified in the $\pi$-calculus up to divergence-preserving branching bisimilarity. The converse, however, is not true due to (intended) limitations of the model of reactive Turing machines. That is, the $\pi$-calculus allows the specification of behaviour that is not finitely executable up to divergence-preserving branching bisimilarity. We shall prove, however, that if the finiteness requirement on reactive Turing machines and the associated notion of executability is relaxed to orbit-finiteness, then the $\pi$-calculus is executable up to (divergence-insensitive) branching bisimilarity.

### 15. Output-sensitive Information flow analysis

Constant-time programming is a countermeasure to prevent cache based attacks where programs should not perform memory accesses that depend on secrets. In some cases this policy can be safely relaxed if one can prove that the program does not leak more information than the public outputs of the computation. We propose a novel approach for verifying constant-time programming based on a new information flow property, called output-sensitive noninterference. Noninterference states that a public observer cannot learn anything about the private data. Since real systems need to intentionally declassify some information, this property is too strong in practice. In order to take into account public outputs we proceed as follows: instead of using complex explicit declassification policies, we partition variables in three sets: input, output and leakage variables. Then, we propose a typing system to statically check that leakage variables do not leak more information about the secret inputs than the public normal output. The novelty of our approach is that we track the dependence of leakage variables with respect not only to the initial values of input variables (as in classical approaches for noninterference), but taking also into account the final values of output variables. We adapted this approach to LLVM IR and we developed a prototype to verify LLVM implementations.

### 16. A Formal Proof of the Irrationality of $\zeta(3)$

This paper presents a complete formal verification of a proof that the evaluation of the Riemann zeta function at 3 is irrational, using the Coq proof assistant. This result was first presented by Apéry in 1978, and the proof we have formalized essentially follows the path of his original presentation. The crux of this proof is to establish that some sequences satisfy a common recurrence. We formally prove this result by an a posteriori verification of calculations performed by computer algebra algorithms in a Maple session. The rest of the proof combines arithmetical ingredients and asymptotic analysis, which we conduct by extending the Mathematical Components libraries.

### 17. Datatype defining rewrite systems for naturals and integers

A datatype defining rewrite system (DDRS) is an algebraic (equational) specification intended to specify a datatype. When interpreting the equations from left-to-right, a DDRS defines a term rewriting system that must be ground-complete. First we define two DDRSs for the ring of integers, each comprising twelve rewrite rules, and prove their ground-completeness. Then we introduce natural number and integer arithmetic specified according to unary view, that is, arithmetic based on a postfix unary append constructor (a form of tallying). Next we specify arithmetic based on two other views: binary and decimal notation. The binary and decimal view have as their characteristic that each normal form resembles common number notation, that is, either a digit, or a string of digits without leading zero, or the negated versions of the latter. Integer arithmetic in binary and decimal notation is based on (postfix) digit append functions. For each view we define a DDRS, and in each case the resulting datatype is a canonical term algebra that extends a corresponding canonical term algebra for natural numbers. Then, for each view, we consider an alternative DDRS based on tree constructors that yields comparable normal forms, which for that view admits expressions that are algorithmically more involved. For all DDRSs considered, ground-completeness is proven.


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Table 4.

Rates of diffusion through alginate-containing Noble agar

DrugDiffusion rate (%)a
1% alginate2% alginate
Levofloxacin60.43 ± 2.2937.07 ± 0.31
Ciprofloxacin60.30 ± 7.0224.07 ± 4.45
Gentamicin<14.9<14.9
Ceftazidime80.83 ± 5.6975.47 ± 0.06
• a Relative to the rate without alginate. Values are means ± standard deviations for three replicate observations, except for gentamicin.


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# Number Properties Puzzle – Challenge 9

Do you enjoy exercising your brain? If so, then this puzzle is just for you. Let's see if you can solve this Math puzzle!

## Challenge:

How many positive integers less than 10,000 are there in which the sum of the digits equals 5?

A- 31

B- 51

C- 56

D- 62

E- 93

$11.99 Satisfied 123 Students The correct answer is C. We are looking for a number that the sum of its digits equals 5. Pairs possible: $$0,0,0,5; 1,4,0,0; 0,0,2,3; 1,3,1,0; 1,2,1,1; 0,1,2,2$$ Number of options for each pair: $$0,0,0,5 = \frac{4!}{3!} = \frac{4 ×3 ×2 ×1}{3 ×2 ×1} = 4 (5, 50, 500, 5,000)$$ $$1,4,0,0 = \frac{4!}{2!} = \frac{4 ×3 ×2 ×1}{2 ×1} = 12 (14, 41, 104, 140, 410, 401, 1004, 1040, 1400, 4001, 4010, 4100)$$ $$0,0,2,3 = \frac{4!}{2!} = \frac{4 ×3 ×2 ×1}{2 ×1} = 12 (23, 32, 230, 320, …)$$ $$1,3,1,0 = \frac{4!}{2!} =\frac{4 ×3 ×2 ×1}{2 ×1)} = 12 (113, 131, 311, 1013, …)$$ $$1,2,1,1 = \frac{4!}{3!} = \frac{4 ×3 ×2 ×1}{3 ×2 ×1} = 4 (1112, 1121, 1211, 2111)$$ $$0,1,2,2 = \frac{4!}{2! }=\frac{4 ×3 ×2 ×1}{2 ×1} = 12 (122, 212, 221, 1022, …)$$ Total: $$12 + 12 + 12 + 12 + 4 + 4 = 56$$ ### More math articles ### What people say about "Number Properties Puzzle - Challenge 9"? No one replied yet. X 30% OFF Limited time only! Save Over 30% SAVE$5

It was $16.99 now it is$11.99


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# Introduction

We use memoir.cls as the main document class for all our books and articles that we typeset. Hence, all the main formatting is done by memoir. The class is available in all standard TeX distributions and can be downloaded from Comprehensive TeX Archive Network (CTAN) from this link. It is well documented and therefore, readers are advised to go through the documentation before making use of this package.

ithal.sty is a subsidiary LaTeX package that will work in conjunction with memoir.cls extending its functionality to meet the typesetting requirements of Malayalam books released by Sayahna Foundation.

The package shall be loaded with the command anywhere before \begin{document}

  \usepackage[<options>]{ithal}

# Options

Various options of the package are given below:

• ml — typeset Malayalam documents.
• print — generate output for printing, default is false.
• proof — proof mode, allows to have line numbers, watermark and footer instruction for proofers.
• sphone — generate pdf output for standard mobile phone with aspect ratio of 16:9.
• mini — generate pdf output for phone with 5 inch screen size.
• kindle — generate pdf output for older versions of Kindle device.

Default is web version pdf with dimensions of 145mm × 195mm, which is fit to read in tablet or iPad. The print version has the paper width × paper height dimensions of 146mm × 210mm; sphone version has the dimensions of 80mm × 140mm; mini has 62mm × 110mm; and lastly kindle has 90mm × 122mm.

Various dimensions like paper width/height, inner/outer margins, head/foot heights, marginpar width/sep and various other page dimension parameters can be changed with key-value options provided as arguments of \geometry command. geometry package is loaded by default.

# Package Dependencies

This package depends on the following LaTeX packages:

1. geometry — controls page geometry and various page dimensions.
2. footmisc — manage and configure footnotes.
3. multicol — provides to typeset text in columnar mode.
4. multirow — needed to span tabular cells across rows.
5. makecell — helps to typeset large chunks of text in a tabular cells and reconfigure its alignment, etc per cell.
6. ifthen — needed internally to manage booleans.
7. enumitem — list management features.
8. ml-draftcopy — add watermarks and other objects in a separate layer in the pdf.
9. etoolbox — needed internally to manage certain programming features.
10. polyglossia — multi-lingual setup.
11. fontspec — font management for utf-8 fonts.
12. xltxtra — provides macros for typesetting XeLaTeX and XeTeX logos, etc.
13. xcolor — color management features.
14. hyperref — hyperlinking internal and external objects.
15. xspace — needed internally for macro managements.


text:
Browse Questions

# What is the ratio of ATP requirement for the fixation of 6 molecules of $CO_2$ in sugarcane and 5 molecules of $N_2$ in bean?

$\begin {array} {1 1} (1)\;5 \: : \: 16 & \quad (2)\;3\: :\: 16 \\ (3)\;5\: :\: 8 & \quad (4)\;3\: :\: 8 \end {array}$

(4) 3 : 8


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# Why does the absence of diabatic heating rate and frictional terms make the zonal and vertical velocity identically zero?

I was reading an article on fluid dynamics of tropical cyclones, where I came across a condition which states that,

For adiabatic friction less flow, equations (1) to (5) have a solution, v(r, z), for a steady freely spinning vortex in which u and w are identically zero and v(r, z) is an arbitrary function of r and z.

Where these 5 equations are basic primitive equations (2 horizontal and 1 vertical equation of motion,conservation of mass and thermodynamic equation) written in cylindrical coordinates.

If I try to consider the horizontal(u component only) and vertical equations of motion when they are in gradient wind and hydrostatic balance, u and w will be conserved i.e du/dt=0 and dw/dt=0. How can one conclude that they are identically zero? For the article, please open the following link https://www.google.co.in/url?sa=t&source=web&rct=j&url=https://www.meteo.physik.uni-muenchen.de/~roger/Publications/M17.pdf&ved=2ahUKEwjvzPOsmOzYAhUeSo8KHf3wC5YQFjAEegQIDRAB&usg=AOvVaw3uxV6Wyli4JcGq3qrQzR00

The primitive equations are given in section 2.1 and the lines highlighted above are taken from the first paragraph of section 2.2

• Please, cite your source. Context matters. For example, are you sure that those u,v,w are not formulated in a locally co-rotating frame? – AtmosphericPrisonEscape Jan 22 '18 at 17:58
• They say they're using a 'steady, freely spinning vortex' so they're already plugging in u and w as zero as initial conditions. Then by requirement u and w stay that way. Plugging that into the equations and checking, shows that there is no contradiction, so $u=w=0$ can be (part of) the solution. – AtmosphericPrisonEscape Jan 22 '18 at 18:57


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# Documentation build failed with Workbench

I'm starting to use Wolfram Workbench 2 to develop a Mathematica package. For the first time I tried "Documentation - Build" under the "Application Tools" but got fatal errors:

BUILD FAILED /Users/[...]/docbuild.xml:132: The following error occurred while executing this line: /Users/[...]/docbuild.xml:90: taskdef A class needed by class com.wolfram.jlink.util.MathematicaTask cannot be found: org/apache/tools/ant/Task

This is with Mathematica 9.0.0.0 on a Mac with OS 10.9.

Any idea on what's wrong here?

-
I would guess that you don't have ant installed properly. –  halirutan Jun 2 '14 at 17:21
Chris, did you try to create documentation and build the NumericalAnalysis package bundled with WB 2.0 ?This package comes with documentation. See if you get the same error with it. If yes, then the problem lies in your WB installation, overwise the problem is perhaps in the way you wrote the documentation. Also: do you get other messages? Normally WB is very verbose, telling you about every step it is doing. Is there anything that went well during the build? –  magma Jun 3 '14 at 13:42
Thanks for the idea. Although the problem fixed itself (see my answer below), I couldn't locate the NumericalAnalysis package you mentioned. –  Chris K Jun 4 '14 at 13:31
@ChrisK the NumericalAnalysis package should be available on the WB site under Examples. You probably have to download it –  magma Jun 5 '14 at 11:26


text:
Physics SIMPLE HARMONIC MOTION, VELOCITY, ACCELERATION AND FORCE LAW FOR SIMPLE HARMONIC MOTION

### Topic Covered

color{blue}{star} SIMPLE HARMONIC MOTION
color{blue}{star} VELOCITY AND ACCELERATION IN SIMPLE HARMONIC MOTION
color{blue}{star} FORCE LAW FOR SIMPLE HARMONIC MOTION

### SIMPLE HARMONIC MOTION

color{blue} ✍ Consider a particle oscillating back and forth about the origin of an x-axis between the limits + A and – A as shown in Fig. 14.3.

color{blue} ✍ This oscillatory motion is said to be simple harmonic if the displacement x of the particle from the origin varies with time as

color{blue} {x (t) = A cos (ω t + φ)}

........................ (14.4)

color{blue} ✍ where A, ω and φ are constants.

color{blue} ✍ Thus, simple harmonic motion (SHM) is not any periodic motion but one in which displacement is a sinusoidal function of time. Fig. 14.4 shows what the positions of a particle executing SHM are at discrete value of time, each interval of time being T/4 where T is the period of motion.

color{blue} ✍Fig. 14.5 plots the graph of x versus t, which gives the values of displacement as a continuous function of time. The quantities A, ω and φ which characterize a given SHM have standard names, as summarised in Fig. 14.6. Let us understand these quantities.

color{blue} ✍The amplitutde A of SHM is the magnitude of maximum displacement of the particle. [Note, A can be taken to be positive without any loss of generality].

color{blue} ✍As the cosine function of time varies from +1 to –1, the displacement varies between the extremes A and – A. Two simple harmonic motions may have same ω and φ but different amplitudes A and B, as shown in Fig. 14.7 (a).

color{blue} ✍While the amplitude A is fixed for a given SHM, the state of motion (position and velocity) of the particle at any time t is determined by the argument (ωt + φ) in the cosine function. This time-dependent quantity, (ωt + φ) is called the phase of the motion.

color{blue} ✍The value of plase at t = 0 is φ and is called the phase constant (or phase angle). If the amplitude is known, φ can be determined from the displacement at t = 0. Two simple harmonic motions may have the same A and ω but different phase angle φ, as shown in Fig. 14.7 (b).

color{blue} ✍Finally, the quantity ω can be seen to be related to the period of motion T. Taking, for simplicity, φ = 0 in Eq. (14.4), we have

color{blue} {x(t ) = A cos ωt}

...................... (14.5)

color{blue} ✍Since the motion has a period T, x(t) is equal to x (t + T). That is,

color{blue} {A cos ωt = A cos ω (t + T )}

.................... (14.6)

color{blue} ✍Now the cosine function is periodic with period 2π, i.e., it first repeats itself when the argument changes by 2π. Therefore,

color{purple} {ω(t + T ) = ωt + 2π}

that is

color{blue} {ω = 2π//T}

....................(14.7)

color{blue} ✍ ω is called the angular frequency of SHM. Its S.I. unit is radians per second. Since the frequency of oscillations is simply 1//T, ω is 2π times the frequency of oscillation.

color{blue} ✍Two simple harmonic motions may have the same A and φ, but different ω, as seen in Fig. 14.8. In this plot the curve (b) has half the period and twice the frequency of the curve (a).

Q 3169680515

Which of the following functions of time represent (a) simple harmonic motion and (b) periodic but not simple harmonic? Give the period for each case.
(1) sin ωt – cos ωt
(2) sin2 ωt
Class 11 Chapter 14 Example 3
Solution:

(a) sin ωt – cos ωt

= sin ωt – sin (π/2 – ωt)

= 2 cos (π/4) sin (ωt – π/4)

= √2 sin (ωt – π/4)

This function represents a simple harmonic motion having a period T = 2π/ω and a phase angle (–π//4) or (7π/4)

(b) sin2 ωt = ½ – ½ cos 2 ωt

The function is periodic having a period T = π/ω. It also represents a harmonic motion with the point of equilibrium occurring at ½ instead of zero.

### SIMPLE HARMONIC MOTION AND UNIFORM CIRCULAR MOTION

color{blue} ✍In this section we show that the projection of uniform circular motion on a diameter of the circle follows simple harmonic motion. A simple experiment (Fig. 14.9) helps us visualize this connection.

color{blue} ✍Tie a ball to the end of a string and make it move in a horizontal plane about a fixed point with a constant angular speed. The ball would then perform a uniform circular motion in the horizontal plane. Observe the ball sideways or from the front, fixing your attention in the plane of motion.

color{blue} ✍The ball will appear to execute to and fro motion along a horizontal line with the point of rotation as the midpoint. You could alternatively observe the shadow of the ball on a wall which is perpendicular to the plane of the circle.

color{blue} ✍In this process what we are observing is the motion of the ball on a diameter of the circle normal to the direction of viewing.

color{blue} ✍Fig. 14.10 describes the same situation mathematically. Suppose a particle P is moving uniformly on a circle of radius A with angular speed ω. The sense of rotation is anticlockwise. The initial position vector of the particle, i.e.,

color{blue} ✍the vector OP at t = 0 makes an angle of φ with the positive direction of x-axis. In time t, it will cover a further angle ωt and its position vector will make an angle of ωt + φ with the +v e xaxis.

color{blue} ✍Next consider the projection of the position vector OP on the x-axis. This will be OP′. The position of P′ on the x-axis, as the particle P moves on the circle, is given by

color{purple} {x(t) = A cos (omega + phi)}

color{blue} ✍which is the defining equation of SHM. This shows that if P moves uniformly on a circle, its projection P′ on a diameter of the circle executes SHM.

color{blue} ✍The particle P and the circle on which it moves are sometimes referred to as the reference particle and the reference circle respectively.

color{blue} ✍We can take projection of the motion of P on any diameter, say the y-axis. In that case, the displacement y(t) of P′ on the y-axis is given by

color{purple} { y =A sin (omega t + phi)}

which is also an SHM of the same amplitude as that of the projection on x-axis, but differing by a phase of π//2.

color{blue} ✍In spite of this connection between circular motion and SHM, the force acting on a particle in linear simple harmonic motion is very different from the centripetal force needed to keep a particle in uniform circular motion.
Q 3189780617

Fig. 14.10 depicts two circular motions. The radius of the circle, the period of revolution, the initial position and the sense of revolution are indicated on the figures. Obtain the simple harmonic motions of the x-projection of the radius vector of the rotating particle P in each case.
Class 11 Chapter 14 Example 4
Solution:

(a) At t = 0, OP makes an angle of 45^o = π/4 rad with the (positive direction of ) x-axis. After time t, it covers an angle (2pi)/T t in the anticlockwise sense, and makes an angle of (2pi)/T + pi/4 with the x -axis.

The projection of OP on the x-axis at time t is given by,

color{ pink} }x (t) = A cos ( (2pi)/T t + pi/4 )}

For T = 4 s,

x(t) = A cos ( (2pi)/4 t + pi/4 )

which is a SHM of amplitude A, period 4 s, and an initial phase = pi/4

(b) In this case at t = 0, OP makes an angle of 90^o = pi/2 with the x -axis. After a time t, it covers an angle of (pi/2 - (2pi/T t ) with the x -axis. The projection of OP on the x-axis at time t is given by

x(t) = B cos(pi/2 - ( 2pi)/T t )

=B sin ((2pi)/T t )

For T =30 s

x(t) = B sin (pi/15 t )

Writing this as x (t ) = B cos ( pi/15 t - pi/2 ) and comparing with Eq. (14.4). We find that this represents a SHM of amplitude B, period 30 s, and an initial phase of - pi/2

### VELOCITY AND ACCELERATION IN SIMPLE HARMONIC MOTION

color{blue} ✍The speed of a particle v in uniform circular motion is its angular speed ω times the radius of the circle A.

color {blue} {v = ω A}

................... (14.8)

color{blue} ✍The direction of velocity v at a time t is along the tangent to the circle at the point where the particle is located at that instant. From the geometry of Fig. 14.11, it is clear that the velocity of the projection particle P′ at time t is

color{blue} {v(t ) = –ωA sin (ωt + φ )}

..................... (14.9)

color{blue} ✍where the negative sign shows that v (t) has a direction opposite to the positive direction of x-axis. Eq. (14.9) gives the instantaneous velocity of a particle executing SHM, where displacement is given by Eq. (14.4). We can, of course, obtain this equation without using geometrical argument, directly by differentiating (Eq. 14.4) with respect of t:

color {blue} { v(t) = d /(dt ) x (t)

...................14.10

color{blue} ✍The method of reference circle can be similarly used for obtaining instantaneous acceleration of a particle undergoing SHM. We know that the centripetal acceleration of a particle P in uniform circular motion has a magnitude v^2//A or ω^2A, and it is directed towards the centre i.e., the direction is along PO. The instantaneous acceleration of the projection particle P′ is then (See Fig. 14.12)

color{purple} { a (t) = –ω^2A cos (ωt + φ) }

color{blue} {= - omega^2 x (t)}

...........................(14.11)

color{blue} ✍Eq. (14.11) gives the acceleration of a particle in SHM. The same equation can again be obtained directly by differentiating velocity v(t) given by Eq. (14.9) with respect to time:

color{blue} {a(t) = d/(dt) v (t)}

.............................(14.12)

color{blue} ✍We note from Eq. (14.11) the important property that acceleration of a particle in SHM is proportional to displacement. For x(t) > 0, a(t) < 0 and for x(t) < 0, a(t) > 0.

color{blue} ✍Thus, whatever the value of x between –A and A, the acceleration a(t) is always directed towards the centre. For simplicity, let us put φ = 0 and write the expression for x (t), v (t)  and a(t) " " x(t) = A cos ωt, v(t) = – ω Asin ωt, a(t)=–ω2 A cos ωt The corresponding plots are shown in Fig. 14.13.

color{blue} ✍All quantitites vary sinusoidally with time; only their maxima differ and the different plots differ in phase. x varies between –A to A; v(t) varies from –ωA to ωA and a(t) from –ω^2A to ω^2A.

color{blue} ✍With respect to displacement plot, velocity plot has a phase difference of π/2 and acceleration plot has a phase difference of π.

Q 3149880713

A body oscillates with SHM according to the equation (in SI units),

x = 5 cos [2π t + π/4].

At t = 1.5 s, calculate the

(a) displacement, (b) speed and (c) acceleration of the body.
Class 11 Chapter 14 Example 5
Solution:

The angular frequency ω of the body = 2π s^-1 and its time period T = 1 s.

At t = 1.5 s

(a) displacement = (5.0 m) cos [(2π s^-1) × 1.5 s + π/4]

= (5.0 m) cos [(3π + π/4)]

= –5.0 × 0.707 m

= –3.535 m

(b) Using Eq. (14.9), the speed of the body

= – (5.0 m)(2π s^-1) sin [(2π s^-1) × 1.5 s + π/4]

= – (5.0 m)(2π s^-1) sin [(3π + π/4)]

= 10π × 0.707 m s^-1

= 22 m s^-1

(c) Using Eq.(14.10), the acceleration of the body

= –(2π s^-1)^2 × "displacement"

= – (2π s^-1)^2 × (–3.535 m)

= 140 m s^-2

### FORCE LAW FOR SIMPLE HARMONIC MOTION

color{blue} ✍Using Newton’s second law of motion, and the expression for acceleration of a particle undergoing SHM (Eq. 14.11), the force acting on a particle of mass m in SHM is

F (t ) = ma

= –mω^2 x (t ).

i.e.,

color{blue} {F (t ) = –k x (t )}

..................... (14.13)

where

color{blue} {k = mω^2}

.............. (14.14a)

or

color{blue} {omega = sqrt (k/m)}

............(14.14b)

color{blue} ✍Like acceleration, force is always directed towards the mean position - hence it is sometimes called the restoring force in SHM.

color{blue} ✍To summarize the discussion so far, simple harmonic motion can be defined in two equivalent ways, either by Eq. (14.4) for displacement or by Eq. (14.13) that gives its force law. Going from Eq. (14.4) to Eq. (14.13) required us to differentiate two times. Likewise by integrating the force law Eq. (14.13) two times, we can get back Eq. (14.4).

color{brown} {"Note"} that the force in Eq. (14.13) is linearly proportional to x(t). A particle oscillating under such a force is, therefore, calling a linear harmonic oscillator. In the real world, the force may contain small additional terms proportional to x^2, x^3, etc. These then are called non-linear oscillators.
Q 3109880718

Two identical springs of spring constant k are attached to a block of mass m and to fixed supports as shown in Fig. 14.14. Show that when the mass is displaced from its equilibrium position on either side, it executes a simple harmonic motion. Find the period of oscillations.
Class 11 Chapter 14 Example 6
Solution:

Let the mass be displaced by a small distance x to the right side of the equilibrium position, as shown in Fig. 14.15. Under this situation the spring on the left side gets elongated by a length equal to x and that on the right side gets compressed by the same length. The forces acting on the mass are then,

color{red} {F_1 = –k x} (force exerted by the spring on the left side, trying to pull themass towards the mean position)

color{red} {F_2 = –k x} (force exerted by the spring on the right side, trying to push the mass towards the mean position)

The net force, F, acting on the mass is then given by,

color{orange} {F = –2kx}

Hence the force acting on the mass is proportional to the displacement and is directed towards the mean position; therefore, the motion executed by the mass is simple harmonic. The time period of oscillations is,

 color{red} {T =2pi sqrt(m/(2k) )}


text:
200701Liang Zhibing

Title: Relative Szemer\'edi theorem for Pseudorandom measures.
Speaker: Liang Zhibin, Peking University.
Time:       Four talks，  Jan 16 and Jan 30, 2007 9:30am-11:30am and 2:00pm-4:00pm

Room: Insititue of Math 509 and 703

Abstracts: My talks is based on section 6-8 of Green-Tao's paper. I will introduce the proof of Szemer\'edi theorem relative to a pseudoramdon measure. It will take about three speeches for me to complete all the proof. In the first speech I will give the definition of the dual norm of Gowers norm and the Gowers anti-uniform functions. In the second speech I will introduce the $\sigma$-algebra and generalised Koopman-von Neumann structure theorem. In the last time I will complete the left of the proof of Szemer\'edi theorem relative to a pseudoramdon measure, using Furstenberg tower.

Photos: To be ADDED In NEXT Time


text:
Somewhat Frequent
0/14

# Flood Fill

Author: Darren Yao

Contributors: Kevin Sheng, George Pong

Finding connected components in a graph represented by a grid.

## Resources

Resources
IUSACO

module is based off this

CP2

code + example

## Introduction

Flood fill is an algorithm that identifies and labels the connected component that a particular cell belongs to in a multidimensional array.

For example, suppose that we want to split the following grid into components of connected cells with the same number.

 2 2 1 2 1 2 2 2 1

Let's start the flood fill from the top-left cell. The color scheme will be red for the node currently being processed, blue for nodes already visited, and uncolored for nodes not yet visited.

 2 2 1 2 1 2 2 2 1
 2 2 1 2 1 2 2 2 1
 2 2 1 2 1 2 2 2 1
 2 2 1 2 1 2 2 2 1
 2 2 1 2 1 2 2 2 1
 2 2 1 2 1 2 2 2 1
 2 2 1 2 1 2 2 2 1
 2 2 1 2 1 2 2 2 1
 2 2 1 2 1 2 2 2 1
 2 2 1 2 1 2 2 2 1

As opposed to an explicit graph where the edges are given, a grid is an implicit graph. This means that the neighbors are just the nodes directly adjacent in the four cardinal directions.

Usually, grids given in problems will be $N$ by $M$, so the first line of the input contains the numbers $N$ and $M$. In this example, we will use a two-dimensional integer array to store the grid, but depending on the problem, a two-dimensional character array or a two-dimensional boolean array may be more appropriate. Then, there are $N$ rows, each with $M$ numbers containing the contents of each square in the grid. Example input might look like the following (varies between problems):

3 4
1 1 2 1
2 3 2 1
1 3 3 3

And we’ll want to input the grid as follows:

C++

#include <iostream>
using namespace std;
const int MAX_N = 1000;
int grid[MAX_N][MAX_N];int row_num;int col_num;

Java

import java.io.*;import java.util.StringTokenizer;
public class Floodfill {	private static int[][] grid;	private static int rowNum;	private static int colNum;	public static void main(String[] args) throws IOException {		BufferedReader read = new BufferedReader(new InputStreamReader(System.in));		StringTokenizer dims = new StringTokenizer(read.readLine());

Python

row_num, col_num = [int(i) for i in input().split()]grid = []for _ in range(row_num):	grid.append([int(i) for i in input().split()])

### Implementation

When doing flood fill, we will maintain an $N\times M$ array of booleans to keep track of which squares have been visited, and a global variable to maintain the size of the current component we are visiting. Make sure to store the grid, the visited array, dimensions, and the current size variable globally.

This means that we want to recursively call the search function for the squares above, below, and to the left and right of our current square. Due to its recursive nature, floodfill can be thought of as a modified version of DFS. The algorithm to find the size of a connected component in a grid using flood fill is as follows (we’ll also maintain a 2D visited array).

The code below shows the global/static variables we need to maintain while doing flood fill and the flood fill algorithm itself:

C++

const int MAX_N = 1000;
int grid[MAX_N][MAX_N];  // the grid itselfint row_num;int col_num;bool visited[MAX_N][MAX_N];  // keeps track of which nodes have been visitedint curr_size = 0;  // reset to 0 each time we start a new component
void floodfill(int r, int c, int color){	if (

Java

public class Floodfill {	private static int[][] grid;  // the grid itself	private static int rowNum;	private static int colNum;  // grid dimensions, rows and columns	private static boolean[][] visited;  // keeps track of which nodes have been visited	private static int currSize = 0;  // reset to 0 each time we start a new component
public static void main(String[] args) {		/*		 * input code and other problem-specific stuff here

Python

import sys
MAX_N = 100sys.setrecursionlimit(2 ** 30)  # pretty much disable the recurion limit
row_num = MAX_Ncol_num = MAX_Ngrid = [[0 for _ in range(col_num)] for _ in range(row_num)]visited = [[False for _ in range(col_num)] for _ in range(row_num)]curr_size = 0

## Example - Counting Rooms

Focus Problem – try your best to solve this problem before continuing!

### Warning!

Recursive implementations of flood fill sometimes lead to

• Stack overflow if you didn't include the appropriate compiler options for adjusting the stack size
• Memory limit exceeded if you run the recursive implementation on a really big grid (i.e., running the above code on a $4000\times 4000$ grid may exceed 256 MB)
• Non-recursive implementations generally use less memory than recursive ones.

A non-recursive implementation of flood fill adds adjacent nodes to a stack or queue, similar to BFS, and is usually implemented as follows:

C++

#include <iostream>#include <string>#include <stack>
using namespace std;
const int MAX_N = 2500;const int R_CHANGE[]{0, 1, 0, -1};const int C_CHANGE[]{1, 0, -1, 0};

Java

import java.io.*;import java.util.*;
public class RoomCount {	public static final int R_CHANGE[] = {0, 1, 0, -1};	public static final int C_CHANGE[] = {1, 0, -1, 0};
private static boolean[][] visited;	private static char[][] building;	private static int rowNum;

Python

row_num, col_num = [int(i) for i in input().split()]grid = [input() for _ in range(row_num)]visited = [[False for _ in range(col_num)] for _ in range(row_num)]

def floodfill(r: int, c: int):	global visited
"""	Note: you can also use queue and pop from the

### Python Optimization: Padding

We can speed up floodfill by a constant time factor using an alternate way of ensuring no out-of-bounds accessing of the grid happens. Imagine a one-cell padding on all four sides of the array, for which the cells of which are marked as already having been visited. We can then get rid of the bound checking where we check if the next cell we visit is inclusive between 0 and the dimensions of the array. These cells in the padding may be added to the queue, but when they are visited in the floodfill loop, they are skipped for being marked visited and no out-of-bounds accessing happens. Furthermore, no additional out-of-bounds cells beyond the padding are added to the queue.

One option for adding padding would be to start our entire grid with a filler row in the beginning, and also start each row with an element before we take in input. After input, then we can add the padding at the ends of rows and bottom of the grid. However, in python we only need to do the latter part as we can have the padding at the right and bottom of the matrix act as the padding for the left and top as the coordinate for the left and top, -1, would be interpreted as the end of the array.

The speed benefit of this optimization is only noticeable for some large inputs (it may be even slower for small inputs) and likely only beneficial when using a slow language such as Python, where even the expected solution may TLE. Our new implementation for the previous problem would be as follows.

row_num, col_num = [int(i) for i in input().split()]
# We make each row a list so we can append at the end.grid = [[char for char in input()] for _ in range(row_num)]visited = [[False for _ in range(col_num)] for _ in range(row_num)]
# We add padding heregrid.append(['#'] * len(grid[0]))  # Add the bottom rowvisited.append([True] * len(visited[0]))for row_grid, row_visited in zip(grid, visited):  # Add right side

## Problems

StatusSourceProblem NameDifficultyTags
SilverEasy
Show TagsFlood Fill
CFEasy
Show TagsFlood Fill
Old SilverEasy
Show TagsBinary Search, Flood Fill
SilverNormal
Show TagsFlood Fill
SilverNormal
Show TagsFlood Fill
SilverNormal
Show TagsFlood Fill
SilverNormal
Show TagsFlood Fill
SilverNormal
Show TagsFlood Fill
SilverNormal
Show TagsFlood Fill
SilverNormal
Show TagsFlood Fill
SilverHard
Show TagsFlood Fill
SilverHard
Show TagsFlood Fill
SilverHard
Show TagsFlood Fill

### Join the USACO Forum!

Stuck on a problem, or don't understand a module? Join the USACO Forum and get help from other competitive programmers!


text:
# Definition:Submersion

## Open Sets in $\R^n$

Let $k, m, n \ge 1$ be natural numbers.

Let $U \subset \R^n$ be open.

Let $f: U \to \R^m$ be a mapping.

Then $f$ is a $C^k$-submersion if and only if $f$ is of class $C^k$ and its differential $\d f$ is surjective at every point of $U$.

### Rank

The rank of a submersion is the rank of its differential at any point.

## Manifolds

Let $X$ and $Y$ be manifolds, with $\dim X \ge \dim Y$.

Let $f: X \to Y$ be smooth and $\map f x = y$.

Let $\d f_x: \map {T_x} X \to \map {T_y} Y$ be a surjection.

Then $f$ is a submersion of $X$ on $Y$.


text:
03.4matrixtransform1

# Transformations with Matrices

We can use matrices to represent and manipulate the transformations we have been performing on graphs.

.

## Notation

If we start with a set of points $(x, \; y)$

Then we transform them using a process, T, to get a new set of points $(x’, \; y’)$
… … (sometimes called the image)

We can show this in matrix notation as:

… … $\left[ \begin{matrix} x' \\ y' \\ \end{matrix} \right] = T \left( \left[ \begin{matrix} x \\ y \\ \end{matrix} \right] \right)$

.

## Reflections and Dilations

Reflections and Dilations are obtained by multiplying $\left[ \begin{matrix} x \\ y \\ \end{matrix} \right]$ by a transformation matrix.

… … $T = \left[ \begin{matrix} 1&0 \\ 0&1 \\ \end{matrix} \right]$ … causes no change at all

This would be written as:

… … $\left[ \begin{matrix} x' \\ y' \\ \end{matrix} \right] = \left[ \begin{matrix} 1&0 \\ 0&1 \\ \end{matrix} \right] \, \left[ \begin{matrix} x \\ y \\ \end{matrix} \right]$

so {by multiplying matrices}

… … $\left[ \begin{matrix} x' \\ y' \\ \end{matrix} \right] = \left[ \begin{matrix} x \\ y \\ \end{matrix} \right]$

or
… … $x' = x$ … and
… … $y' = y$ … … {ie no change}

• Remember that the matrix shown here is called the Identity Matrix (or I)
• For any matrix, A, it follows from the definition of I that I × A = A

.

## Reflections

.
… … $\left[ \begin{matrix} -1&0 \\ 0&1 \\ \end{matrix} \right]$ … causes a reflection in the x direction (across the y-axis or on the y-axis)
.
… … $\left[ \begin{matrix} 1&0 \\ 0&-1 \\ \end{matrix} \right]$ … causes a reflection in the y direction (across the x-axis or on the x-axis)}

.

A reflection in the x-direction (across the y-axis or on the y-axis) could be written as:

… … $\left[ \begin{matrix} x' \\ y' \\ \end{matrix} \right] = \left[ \begin{matrix} -1&0 \\ 0&1 \\ \end{matrix} \right] \, \left[ \begin{matrix} x \\ y \\ \end{matrix} \right]$
.

so {by multiplying matrices}

… … $\left[ \begin{matrix} x' \\ y' \\ \end{matrix} \right] = \left[ \begin{matrix} -x \\ y \\ \end{matrix} \right]$

or
… … $x' = -x$ … and
… … $y' = y$

For example, if we start with the point $(5, \; 3)$

under this transformation, the image is $(-5, \; 3)$
.

Compare this to starting with the standard cubic graph of $y = x^3$

under this transformation, the image is $y' = (-x')^3$

.

## Dilations

.
… … $\left[ \begin{matrix} a&0 \\ 0&1 \\ \end{matrix} \right]$ … causes a dilation by a factor of a in the x direction (from the y-axis)
.
… … $\left[ \begin{matrix} 1&0 \\ 0&a \\ \end{matrix} \right]$ … causes a dilation by a factor of a in the y direction (from the x-axis) }

.

For example:
… … $\left[ \begin{matrix} x' \\ y' \\ \end{matrix} \right] = \left[ \begin{matrix} 3&0 \\ 0&1 \\ \end{matrix} \right] \left[ \begin{matrix} x \\ y \\ \end{matrix} \right]$
.
… … $\left[ \begin{matrix} x' \\ y' \\ \end{matrix} \right] = \left[ \begin{matrix} 3x \\ y \\ \end{matrix} \right]$
.
This represents a dilation by a factor of 3 in the x direction (all x-values are multiplied by 3).
.

This transformation could be written as:

… … $T \left( \left[ \begin{matrix} x \\ y \\ \end{matrix} \right] \right) = \left[ \begin{matrix} 3x \\ y \\ \end{matrix} \right]$

or
… … $x' = 3x$ .. and
… … $y' = y$

For example, if we start with the point $(5, \; 3)$

under this transformation, the image is $(15, \; 3)$
.

Compare this to starting with the cubic graph $y = x^3$

under this transformation, the image is: $y' = \big( \frac{1}{3}x' \big)^3$
.

• Remember that $y = f(nx)$ causes a dilation by a factor of $\frac{1}{n}$ in the x direction (from the y-axis)}

.

Note:
We can arrive at the new equation alegbraically by making x the subject and substituting:

… … $y' = y$ … and
… … $y = x^3$ so:

… … $y' = x^3$

but
… … $x = \frac{1}{3}x'$

so:
… … $y' = \big( \frac{1}{3}x' \big)^3$

.

In function notation, this transformation could be written as:

… … $\big\{ T: R^2 \rightarrow R^2, \;\; T(x,\;y) \rightarrow (3x,\; y) \big\}$

or in shorthand as:

… … $(x',\;y') \rightarrow (3x,\; y)$
.

• Read the arrow as "maps onto".
• Don't confuse it with the arrow used in limits which means "approaches the limit".
• The domain and co-domain of R2 is because we are working in 2 dimensions (x, y)

.

## Combining Dilations and Reflections

These operations can be combined to cause more than one transformation.

For example, the transformation matrix $\left[ \begin{matrix} -2&0 \\ 0&3 \\ \end{matrix} \right]$ gives:

… … $\left[ \begin{matrix} x' \\ y' \\ \end{matrix} \right] = \left[ \begin{matrix} -2&0 \\ 0&3 \\ \end{matrix} \right] \, \left[ \begin{matrix} x \\ y \\ \end{matrix} \right]$

… … $\left[ \begin{matrix} x' \\ y' \\ \end{matrix} \right] = \left[ \begin{matrix} -2x \\ 3y \\ \end{matrix} \right]$

or

… … $T \left( \left[ \begin{matrix} x \\ y \\ \end{matrix} \right] \right) = \left[ \begin{matrix} -2x \\ 3y \\ \end{matrix} \right]$

or
… … $x' = -2x$ … and
… … $y' = 3y$

This represents

• a reflection in the x direction (across the y-axis) and
• a dilation by a factor of 2 in the x direction (away from the y-axis) and
• a dilation by a factor of 3 in the y direction (away from the x-axis).

For example, if we start with the point $(5, \; 3)$

under this transformation, the image is $(-10, \; 9)$
.

Compare this to starting with a standard cubic graph $y = x^3$

under this transformation, the image is: $y'=3 \big( -\frac{1}{2}x' \big)^3$

.

We can arrive at the new equation algebraically:

… … $y' = 3y$ … and
… … $y = x^3$
so:

… … $y' = 3x^3$

make x the subject of $x' = -2x$ gives us:

… … $x = -\frac{1}{2}x'$

so:

… … $y' = 3\big( -\frac{1}{2}x' \big)^3$

This equation could be simplified by moving the $-\frac{1}{2}$ out of the cube, so we get:

… … $y' = 3 \big( -\frac{1}{2} \big)^3 \big( x' \big)^3$

… … $y' = -\frac{3}{8} \big( x' \big)^3$

.

In function notation, this transformation could be written as:

… … $\big\{ T: R^2 \rightarrow R^2, \;\; T(x, \; y) \rightarrow (-2x, \; 3y) \big\}$
.

or in shorthand as

… … $(x', \; y') \rightarrow (-2x, \; 3y)$
.

• Read the arrow as “maps onto”.
• The domain of R2 is because we are working in 2 dimensions (x, y)

.

.


text:
## WeBWorK Problems

### Checking a basis consisting of row vectors

by Christopher Heckman -
Number of replies: 0
I have decided to revamp a few of the WeBWorK problems for MAT 242 (Elementary Linear Algebra) at Arizona State, and I am currently stuck on the following:

Currently, the problems that ask for a basis for the row space of a matrix A require students to enter the basis as column vectors. What I want to do is to to change this to entry via row vectors. I have the formatting and entry taken care of, but the problem seems to be that basis_cmp can't work with row vectors. Is there some clever hack which will enable it to work?

Here is the code that I have (once the matrix A has been created):

BEGIN_TEXT
Here, x = $x.$BR
Find a basis of the row space of the matrix
$BR \{ mbox( '$$A=$$', display_matrix($A), '.' ) \}
$BR If there are extra vectors which you do not need, fill in their entries with x's.$BR
\{ mbox( ans_array(1,5,10), ', ') \}
\{ mbox( ans_array_extension(1,5,10), ', ') \}
\{ mbox( ans_array_extension(1,5,10), ', ') \}
\{ mbox( ans_array(1,5,10), '. ') \}
END_TEXT

if ($x == 3) { ANS(basis_cmp([[1,0,0,($a[1][4]*$a[2][2]-$a[1][2]*$a[2][4])/($a[2][2]*$a[1][1]), ($a[1][5]*$a[2][2]-$a[1][2]*$a[2][5])/($a[2][2]*$a[1][1])], [0,1,0,$a[2][4]/$a[2][2],$a[2][5]/$a[2][2]], [0,0,1,0,0]])); #$x = 3
} else { if ($x == 4) { ANS(basis_cmp([[1,0,($a[1][3]*$a[2][2]-$a[1][2]*$a[2][3])/($a[2][2]*$a[1][1]),0, ($a[1][5]*$a[2][2]-$a[1][2]*$a[2][5])/($a[2][2]*$a[1][1])], [0,1,$a[2][3]/$a[2][2],0,$a[2][5]/$a[2][2]], [0,0,0,1,0]])); #$x = 4
} else { if ($x == 5) { ANS(basis_cmp([[1,0,($a[1][3]*$a[2][2]-$a[1][2]*$a[2][3])/($a[2][2]*$a[1][1]), ($a[1][4]*$a[2][2]-$a[1][2]*$a[2][4])/($a[2][2]*$a[1][1]),0], [0,1,$a[2][3]/$a[2][2],$a[2][4]/$a[2][2],0], [0,0,0,0,1]])); #$x = 5
} } }
ANS(str_cmp("x")); ANS(str_cmp("x")); ANS(str_cmp("x")); ANS(str_cmp("x"));

ENDDOCUMENT();

(I am currently trying this with a seed which makes \$x = 5. The formula for that answer is correct; I checked it with Maple.)

--- Christopher Heckman

P.S. The extra ans_array and ANS's are intentional; I also wanted to "hide" what the dimensions of these spaces are. Furthermore, I'm changing a few formulas so that the "correct answers" that WeBWorK gives are the ones you get by doing the procedures described in the book; not all the problems have been written with this in mind.


text:
TITLE

# LP-Based Covering Games with Low Price of Anarchy

AUTHOR(S)
Piliouras, Georgios; Valla, Tomáš; Végh, László
PUB. DATE
July 2015
SOURCE
Theory of Computing Systems;Jul2015, Vol. 57 Issue 1, p238
SOURCE TYPE
DOC. TYPE
Article
ABSTRACT
We design a new class of vertex and set cover games, where the price of anarchy bounds match the best known constant factor approximation guarantees for the centralized optimization problems for linear and also for submodular costs. This is in contrast to all previously studied covering games, where the price of anarchy grows linearly with the size of the game. Both the game design and the price of anarchy results are based on structural properties of the linear programming relaxations. For linear costs we also exhibit simple best response dynamics that converge to Nash equilibria in linear time.
ACCESSION #
103167799

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# KJCP3 - Editorial

Take Turns

Author: Dipen Ved
Tester: Dipen Ved
Editorialist: Nirali

EASY

# PREREQUISITES:

Use of iterators and sets

# PROBLEM:

Given two lists, you have to sort them and print them alternatively starting with list one.

# QUICK EXPLANATION:

The input is taken using two 'sets' and two 'for loops' for two lists. Take two iterators, each pointing to the beginning of the respective sets. Using a for loop, alternatively print the numbers the two iterators are pointing to.

# EXPLANATION:

Let's first consider solving the problem for a single test case. We are given two lists. We have to take the input using set so that they are already sorted and we don't have to write a separate code again for sorting the list. So, using for loop and insert function -

for(int i=0;i<n;i++) {
int temp;
scanf("%d",&temp);
set1.insert(temp);
}

Here, n is the size of the list and temp is the number in the list Similarly, for the second list. Now, we wish to print the elements of the lists alternatively starting with list one. So let's put an iterator to point to the beginning to set2.

set<int>::iterator it2 = set2.begin();

We also want an iterator pointing to the start of set1, which we will do in the for loop.

for(set <int>::iterator it1 = set1.begin();it1!=set1.end() && it2!=set2.end();it1++,it2++) {
cout<<*it1<<" "<<*it2<<" ";
}

The for loop will run till both the iterators reach the end of their respective sets and alternatively both sets will be printed.

# AUTHOR'S AND TESTER'S SOLUTIONS:

Solution can be found here

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# How to Reduce Phase Lag Caused by Kalman Filter

Background

I have been developing a system using a moving robot with a distance sensor against another robot. I want to control these robots by estimating relative velocity and acceleration derived from the distance sensor. So I'd like to estimate relative velocity and acceleration using the measured value by the distance sensor. (Due to some restrictions, no additional sensor is allowed.)

Problem statement

I formulated the problem with a cubic Kalman filter configured as follows [1]: \begin{align} \left[ \begin{array}{c} x_k \\ \dot{x}_k \\ \ddot{x}_k \end{array} \right] = \left[ \begin{array}{ccc} 1 && dt && dt^2/2 \\ 0 && 1 && dt \\ 0 && 0 && 1 \end{array} \right] \left[ \begin{array}{c} x_{k-1} \\ \dot{x}_{k-1} \\ \ddot{x}_{k-1} \end{array} \right] + \left[ \begin{array}{c} dt^3/6 \\ dt^2/2 \\ dt \end{array} \right] w_k, \end{align} $$$$z_k = x_k + v_k,$$$$ where process noise and observation noise follow gaussian distributions $$w_k\sim\mathcal{N}(0,Q), v_k\sim\mathcal{N}(0,R)$$.

I modeled $$Q$$ and $$R$$ as below. Q=\begin{align} q\left[ \begin{array}{ccc} dt^5/20 && dt^4/8 && dt^3/6 \\ dt^4/8 && dt^3/3 && dt^2/2 \\ dt^3/6 && dt^2/2 && dt \end{array} \right] \end{align}, R=\frac{e_r^2/4+2r}{3}

Filtered signals have reasonable shapes as illustrated in the figure below but there exists large phase lag caused by the filter. It is not acceptable for realtime control of the robots in reaction to another robot.

I’m seeking a way to reduce the lag caused by the filter.

What I have tried

• Tuning parameters of the filter
Even with best parameters, I observed much phase lag.
• Quadraric Kalman filter applied successively to get acceleration (first Kalman filter is applied to get velocity and another Kalman filter is applied to the estimated velocity to get acceleration.)
Not much difference observed.
• Kalman smoother as a fixed-lag smoother
Not much difference observed.

Demonstration

I simulated the phase lag induced by the Kalman filter for illustration. I generated signals for simluation as follows:

Hz = 1000; % signal's frequency
time = 0:1/Hz:30; % time interval for simulation
accel = 200-400*cos(time*2*pi/40) - 500*sin(time/40) + 80*sin(time/10) .* cos(time/2-10) - 240*cos(time); % true acceleration
vel = cumtrapz(accel)/Hz; % velocity
dist = cumtrapz(vel)/Hz + 10*randn(1,numel(time)); % true distance with gaussian disturbance

Parameters used in the filter: $$q = 20,~ er=1,~ r=10$$. Blue lines show true distance/velocity/acceleration and red lines show filtered distance/velocity/acceleration. In acceleration estimation, we can observe large phase lag.

Any help would be warmly welcomed and appreciated.

EDIT@3/23/2020, UPDATE@3/25/2020

In response to a nice answer by @Luezoid, I also tried a constant jerk model instead of constant acceleration model [2]. The state transition matrix $$F$$ and covariance matrix of the process noise $$Q$$ is changed as follows: \begin{align} F&= \left[ \begin{array}{cccc} 1 && dt && dt^2/2 && dt^3/6 \\ 0 && 1 && dt && dt^2/2 \\ 0 && 0 && 1 && dt \\ 0 && 0 && 0 && 1 \\ \end{array} \right] ,\\ Q&=\left[ \begin{array}{cccc} dt^7/252 && dt^6/72 && dt^5/30 && dt^4/24 \\ dt^6/72 && dt^5/20 && dt^4/8 && dt^3/6 \\ dt^5/30 && dt^4/8 && dt^3/3 && dt^2/2 \\ dt^4/24 && dt^3/6 && dt^2/2 && dt \\ \end{array} \right], \end{align} where we assume time interval is sufficiently small. The other things remain the same as the constant acceleration model.

Here is the code.

x = dist; % dist: simulated signal above

dt = 1/Hz;

F = [1 dt dt^2/2 dt^3/6; 0 1 dt dt^2/2; 0 0 1 dt; 0 0 0 1]; % State trans matrix
H = [1 0 0 0]; % Observation matrix
Q = q * [dt^7/252 dt^6/72 dt^5/30 dt^4/24;...
dt^6/72  dt^5/20 dt^4/8  dt^3/6;...
dt^5/30  dt^4/8  dt^3/3  dt^2/2;...
dt^4/24  dt^3/6  dt^2/2  dt]; % Covariance of process noise
R = (er^2/4+2*r)/3; % Covariance of measurement noise

x_est = [x(1);0;0;0]; % Filtered signal

P = eye(4,4); % Assuming initial estimate is correct

x_filtered = zeros(4,numel(x));
x_filtered(:,1) = x_est;

for t = 2:numel(x)
z = x(t);

x_est = F*x_est;  % Predicted State Estimate
P = F*P*F' + Q;     % Predicted Error Covariance

y = z - H*x_est;     % Innovation or Measurement Pre-fit Residual
S = R + H*P*H';     % Innovation or Pre-fit Residual Covariance

K = P*H'/S;         % Optimal Kalman Gain

x_est = x_est + K*y;      % Updated State Estimate
P = (eye(4,4) - K*H)*P;   % Updated Estimate Covariance

x_filtered(:,t) = x_est;
end

I ran the simulation with the same data and obtained the results with different filter parameters.

Result 1 Constant jerk model with parameters: $$q = 20,~ er=1,~ r=10$$. Result 2 Constant jerk model with parameters: $$q = 50,~ er=1,~ r=10$$.

As observed in the results, I got larger amount of estimation error in acceleration compared to the constant acceleration model. Can we somehow reduce phase lag while keeping error level low?

[1]: L.J.Puglisi et al., On the velocity and Acceleration Estimation from Discrete Time-Position Sensors, CEAI, 2015.

[2]: K. Mehrotra and P. R. Mahapatra, A jerk model for tracking highly maneuvering targets, IEEE Transactions on Aerospace and Electronic Systems, vol. 33, no. 4, pp. 1094-1105, Oct. 1997.

• Hi! You have about one second lag in acceleration estimation. Is it this lag you complain about ? Yet your position estimation seems quite accurate ? (may be still not enough for your application accuracy ?) Mar 9 '20 at 22:36
• @Fat32 Thanks for your comment. Yes the lag in acceleration is too large to build a control system that takes acceleration into account. Mar 10 '20 at 1:16
• what happens if you increase sampling rate ? (decrease sampling period ?) Mar 10 '20 at 2:04
• @Fat32 I ran a simulation with a higher sampling rate and found it reduced the lag, but unfortunately the sampling rate of the distance sensor has already been set to its maximum. Mar 10 '20 at 2:36
• Could you share more of the code? Maybe we'll spot something before writing an answer.
– Royi
Mar 23 '20 at 12:46

What you're experiencing is the transient lag of the Kalman Filter.
The Kalman Filter, using the Measurement and Process Noise balances between begin very adaptive to being an aggressive smoother.

In your case it means either having short lag with high error in steady state or having long lag with small error in steady state.
In other words, either have low error on transients yet larger error on steady state or higher error on transients and lower error on steady state.

One way to deal with such issue is using Interacting Multiple Model (IMM) Kalman Filters where you have multiple Kalman Models and you jump from one to another according to need.

Another, different approach, is using FIR Filtering.
Which doesn't have the infinite horizon (Memory) of the Kalman Filter.

You may read the work by Yuriy S. Shmaliy. Specifically the paper Unbiased FIR Filtering: An Iterative Alternative to Kalman Filtering Ignoring Noise and Initial Conditions (PDF version on ResearchGate).

With finite horizon of an FIR Filter (Basically derived by Least Squares) you may have higher RMSE in theory (Since the Kalman Filter is optimal for the case the model is the same for infinite samples) but in practice you get good results with better balance of the transients and sensitivity to initialization (Which is the other side of the transients sensitivity).

Resources on Optimal FIR:

Have you considered trying a constant jerk model as opposed to a constant acceleration model? Perhaps a higher order model would capture the acceleration better. See, for instance:

K. Mehrotra and P. R. Mahapatra, "A jerk model for tracking highly maneuvering targets," in IEEE Transactions on Aerospace and Electronic Systems, vol. 33, no. 4, pp. 1094-1105, Oct. 1997.

• Thanks for your answer! I tried the constant jerk model (4-state Kalman filter) and obtained less phase-lag acceleration estimation, but also introduced much larger error. I edit the question to share the trial. Can we somehow reduce phase lag while maintaining error level? Mar 23 '20 at 8:25
• Just some thoughts... The simulated acceleration is periodic, so really you would need an infinite order polynomial to truly capture the process dynamics. The process noise captures this model mismatch, so increasing Q essentially gives the model more flexibility, but will increase the estimate noise. Decreasing Q increases lag. An excellent discussion on this topic which directly addresses tradeoffs in model order, selection of Q, and the resulting filter lag and performance can be found in Ch. 5 of Fundamentals of Kalman Filtering by Paul Zarchan. Highly recommend. Mar 23 '20 at 15:34