In mathematics, a function on the real numbers is called a step function if it can be written as a finite linear combination of indicator functions of intervals. Informally speaking, a function is a piecewise constant function having only finitely many pieces. Indeed, if that is not the case to start with, a constant function is a trivial example of a step function. Then there is only one interval, A0 = R, the sign function sgn , which is −1 for negative numbers and +1 for positive numbers, and is the simplest non-constant step function. The Heaviside function H, which is 0 for negative numbers and 1 for positive numbers, is an important step function and it is the mathematical concept behind some test signals, such as those used to determine the step response of a dynamical system. The rectangular function, the boxcar function, is the next simplest step function. The integer part function is not a step function according to the definition of this article, some authors define step functions with an infinite number of intervals.
The sum and product of two functions is again a step function. The product of a function with a number is a step function. As such, the functions form an algebra over the real numbers. A step function takes only a number of values. If the intervals A i, i =0,1, …, n, in the definition of the step function are disjoint and their union is the real line. The definite integral of a function is a piecewise linear function. In fact, this equality can be the first step in constructing the Lebesgue integral, unit step function Crenel function Simple function Piecewise defined function Sigmoid function Step detection
A splay tree is a self-adjusting binary search tree with the additional property that recently accessed elements are quick to access again. It performs basic operations such as insertion, look-up and removal in O amortized time, for many sequences of non-random operations, splay trees perform better than other search trees, even when the specific pattern of the sequence is unknown. The splay tree was invented by Daniel Sleator and Robert Tarjan in 1985, all normal operations on a binary search tree are combined with one basic operation, called splaying. Splaying the tree for a certain element rearranges the tree so that the element is placed at the root of the tree. One way to do this is to first perform a binary tree search for the element in question. Alternatively, an algorithm can combine the search and the tree reorganization into a single phase. Good performance for a tree depends on the fact that it is self-optimizing. The worst-case height—though unlikely—is O, with the average being O, having frequently used nodes near the root is an advantage for many practical applications, and is particularly useful for implementing caches and garbage collection algorithms.
Advantages include, Comparable performance, Average-case performance is as efficient as other trees, small memory footprint, Splay trees do not need to store any bookkeeping data. The most significant disadvantage of splay trees is that the height of a tree can be linear. For example, this will be the case after accessing all n elements in non-decreasing order, since the height of a tree corresponds to the worst-case access time, this means that the actual cost of an operation can be high. However the amortized access cost of this worst case is logarithmic, O. Also, the representation of splay trees can change even when they are accessed in a read-only manner. This complicates the use of such trees in a multi-threaded environment. Specifically, extra management is needed if multiple threads are allowed to perform find operations concurrently and this makes them unsuitable for general use in purely functional programming, although even there they can be used in limited ways to implement priority queues.
When a node x is accessed, an operation is performed on x to move it to the root. To perform a splay operation we carry out a sequence of splay steps and it is important to remember to set gg to now point to x after any splay operation. If gg is null, x obviously is now the root, There are three types of splay steps, each of which has a left- and right-handed case. For the sake of brevity, only one of two is shown for each type
The primary goal of corporate finance is to maximize or increase shareholder value. Investment analysis is concerned with the setting of criteria about which value-adding projects should receive investment funding, the terms corporate finance and corporate financier are associated with investment banking. The typical role of an investment bank is to evaluate the financial needs. Thus, the corporate finance and corporate financier may be associated with transactions in which capital is raised in order to create, develop. Financial management overlaps with the function of the Accounting profession. The primary goal of management is to maximize or to continually increase shareholder value. Managers of growth companies will use most of the capital resources. When companies reach maturity levels within their industry, managers of companies will use surplus cash to payout dividends to shareholders. Choosing between investment projects will be based upon several inter-related criteria, Corporate management seeks to maximize the value of the firm by investing in projects which yield a positive net present value when valued using an appropriate discount rate in consideration of risk.
These projects must be financed appropriately, if no growth is possible by the company and excess cash surplus is not needed to the firm, financial theory suggests that management should return some or all of the excess cash to shareholders. This capital budgeting is the planning of value-adding, long-term corporate financial projects relating to investments funded through, Management must allocate the firms limited resources between competing opportunities. Investments should be made on the basis of value-added to the future of the corporation, projects that increase a firms value may include a wide variety of different types of investments, including but not limited to, expansion policies, or mergers and acquisitions. Achieving the goals of corporate finance requires that any corporate investment be financed appropriately, the sources of financing are, capital self-generated by the firm and capital from external funders, obtained by issuing new debt and equity. As above, since both hurdle rate and cash flows will be affected, the mix will impact the valuation of the firm.
Financing a project through debt results in a liability or obligation that must be serviced, equity financing is less risky with respect to cash flow commitments, but results in a dilution of share ownership and earnings. Management must attempt to match the long-term financing mix to the assets being financed as closely as possible, other techniques, such as securitization, or hedging using interest rate- or credit derivatives, are common. See Asset liability management, Treasury management, Credit risk, Interest rate risk, however economists have developed a set of alternative theories about how managers allocate a corporations finances. Also, Capital structure substitution theory hypothesizes that management manipulates the capital such that earnings per share are maximized
Monte Carlo method
Monte Carlo methods are a broad class of computational algorithms that rely on repeated random sampling to obtain numerical results. Their essential idea is using randomness to solve problems that might be deterministic in principle and they are often used in physical and mathematical problems and are most useful when it is difficult or impossible to use other approaches. Monte Carlo methods are used in three distinct problem classes, numerical integration, and generating draws from a probability distribution. In principle, Monte Carlo methods can be used to any problem having a probabilistic interpretation. By the law of numbers, integrals described by the expected value of some random variable can be approximated by taking the empirical mean of independent samples of the variable. When the probability distribution of the variable is parametrized, mathematicians often use a Markov Chain Monte Carlo sampler, the central idea is to design a judicious Markov chain model with a prescribed stationary probability distribution.
That is, in the limit, the samples being generated by the MCMC method will be samples from the desired distribution, by the ergodic theorem, the stationary distribution is approximated by the empirical measures of the random states of the MCMC sampler. In other problems, the objective is generating draws from a sequence of probability distributions satisfying a nonlinear evolution equation, in other instances we are given a flow of probability distributions with an increasing level of sampling complexity. These models can be seen as the evolution of the law of the states of a nonlinear Markov chain. In contrast with traditional Monte Carlo and Markov chain Monte Carlo methodologies these mean field particle techniques rely on sequential interacting samples, the terminology mean field reflects the fact that each of the samples interacts with the empirical measures of the process. Monte Carlo methods vary, but tend to follow a particular pattern, generate inputs randomly from a probability distribution over the domain.
Perform a deterministic computation on the inputs, for example, consider a circle inscribed in a unit square. Given that the circle and the square have a ratio of areas that is π/4, uniformly scatter objects of uniform size over the square. Count the number of objects inside the circle and the number of objects. The ratio of the two counts is an estimate of the ratio of the two areas, which is π/4, multiply the result by 4 to estimate π. In this procedure the domain of inputs is the square that circumscribes our circle and we generate random inputs by scattering grains over the square perform a computation on each input. Finally, we aggregate the results to obtain our final result, there are two important points to consider here, Firstly, if the grains are not uniformly distributed, our approximation will be poor. Secondly, there should be a number of inputs
In statistics, a confidence interval is a type of interval estimate of a population parameter. It is an interval, in principle different from sample to sample. How frequently the observed interval contains the true parameter if the experiment is repeated is called the confidence level, whereas two-sided confidence limits form a confidence interval, and one-sided limits are referred to as lower/upper confidence bounds. Confidence intervals consist of a range of values that act as good estimates of the population parameter. However, the interval computed from a sample does not necessarily include the true value of the parameter. After any particular sample is taken, the parameter is either in the interval or not. Since the observed data are random samples from the true population, the 99% confidence level means that 99% of the intervals obtained from such samples will contain the true parameter. The desired level of confidence is set by the researcher, If a corresponding hypothesis test is performed, the confidence level is the complement of the level of significance, i. e. a 95% confidence interval reflects a significance level of 0.05.
The confidence interval contains the values that, when tested. Confidence intervals of difference parameters not containing 0 imply that there is a significant difference between the populations. In applied practice, confidence intervals are typically stated at the 95% confidence level, when presented graphically, confidence intervals can be shown at several confidence levels, for example, 90%, 95%, and 99%. Factors affecting the width of the confidence interval include the size of the sample, the level. A larger sample size normally will lead to an estimate of the population parameter. Confidence intervals were introduced to statistics by Jerzy Neyman in a paper published in 1937, Interval estimates can be contrasted with point estimates. A point estimate is a value given as the estimate of a population parameter that is of interest, for example. An interval estimate specifies instead a range within which the parameter is estimated to lie, Confidence intervals are commonly reported in tables or graphs along with point estimates of the same parameters, to show the reliability of the estimates.
For example, an interval can be used to describe how reliable survey results are. In a poll of election–voting intentions, the result might be that 40% of respondents intend to vote for a certain party, a 99% confidence interval for the proportion in the whole population having the same intention on the survey might be 30% to 50%
In computer science, arranging in an ordered sequence is called sorting. Sorting is an operation in many applications, and efficient algorithms to perform it have been developed. The most common uses of sorted sequences are, making lookup or search efficient, the opposite of sorting, rearranging a sequence of items in a random or meaningless order, is called shuffling. For sorting, either a weak order, should not come after, can be specified, or a weak order. For the sorting to be unique, these two are restricted to an order and a strict total order, respectively. Sorting n-tuples can be based on one or more of its components. More generally objects can be sorted based on a property, such a component or property is called a sort key. For example, the items are books, the key is the title, subject or author. A new sort key can be created from two or more sort keys by lexicographical order, the first is called the primary sort key, the second the secondary sort key, etc. For example, addresses could be sorted using the city as primary sort key, if the sort key values are totally ordered, the sort key defines a weak order of the items, items with the same sort key are equivalent with respect to sorting.
If different items have different sort key values this defines an order of the items. A standard order is called ascending, the reverse order descending. For dates and times, ascending means that earlier values precede ones e. g. 1/1/2000 will sort ahead of 1/1/2001, bubble sort, Exchange two adjacent elements if they are out of order. Insertion sort, Scan successive elements for an item, insert the item in the proper place. Selection sort, Find the smallest element in the array, swap it with the value in the first position. Quick sort, Partition the array into two segments, in the first segment, all elements are less than or equal to the pivot value. In the second segment, all elements are greater than or equal to the pivot value, sort the two segments recursively. Merge sort, Divide the list of elements in two parts, sort the two individually and merge it
Time is the indefinite continued progress of existence and events that occur in apparently irreversible succession from the past through the present to the future. Time is often referred to as the dimension, along with the three spatial dimensions. Time has long been an important subject of study in religion and science, diverse fields such as business, sports, the sciences, and the performing arts all incorporate some notion of time into their respective measuring systems. Two contrasting viewpoints on time divide prominent philosophers, one view is that time is part of the fundamental structure of the universe—a dimension independent of events, in which events occur in sequence. Isaac Newton subscribed to this realist view, and hence it is referred to as Newtonian time. This second view, in the tradition of Gottfried Leibniz and Immanuel Kant, holds that time is neither an event nor a thing, Time in physics is unambiguously operationally defined as what a clock reads. Time is one of the seven fundamental physical quantities in both the International System of Units and International System of Quantities, Time is used to define other quantities—such as velocity—so defining time in terms of such quantities would result in circularity of definition.
The operational definition leaves aside the question there is something called time, apart from the counting activity just mentioned, that flows. Investigations of a single continuum called spacetime bring questions about space into questions about time, questions that have their roots in the works of early students of natural philosophy. Furthermore, it may be there is a subjective component to time. Temporal measurement has occupied scientists and technologists, and was a motivation in navigation. Periodic events and periodic motion have long served as standards for units of time, examples include the apparent motion of the sun across the sky, the phases of the moon, the swing of a pendulum, and the beat of a heart. Currently, the unit of time, the second, is defined by measuring the electronic transition frequency of caesium atoms. Time is of significant social importance, having economic value as well as value, due to an awareness of the limited time in each day. In day-to-day life, the clock is consulted for periods less than a day whereas the calendar is consulted for periods longer than a day, personal electronic devices display both calendars and clocks simultaneously.
The number that marks the occurrence of an event as to hour or date is obtained by counting from a fiducial epoch—a central reference point. Artifacts from the Paleolithic suggest that the moon was used to time as early as 6,000 years ago. Lunar calendars were among the first to appear, either 12 or 13 lunar months, without intercalation to add days or months to some years, seasons quickly drift in a calendar based solely on twelve lunar months
In probability theory and related fields, a Markov process, named after the Russian mathematician Andrey Markov, is a stochastic process that satisfies the Markov property. e. Conditional on the present state of the system, its future, a Markov chain is a type of Markov process that has either discrete state space or discrete index set, but the precise definition of a Markov chain varies. Andrey Markov studied Markov processes in the early 20th century, publishing his first paper on the topic in 1906, random walks on the integers and the Gamblers ruin problem are examples of Markov processes and were studied hundreds of years earlier. These two processes are Markov processes in time, while random walks on the integers and the Gamblers ruin problem are examples of Markov processes in discrete time. The algorithm known as PageRank, which was proposed for the internet search engine Google, is based on a Markov process. The adjective Markovian is used to something that is related to a Markov process.
A Markov chain is a process with the Markov property. The term Markov chain refers to the sequence of variables such a process moves through. It can thus be used for describing systems that follow a chain of linked events, the systems state space and time parameter index need to be specified. In addition, there are extensions of Markov processes that are referred to as such. Moreover, the index need not necessarily be real-valued, like with the state space. Notice that the state space continuous-time Markov chain is general to such a degree that it has no designated term. While the time parameter is usually discrete, the space of a Markov chain does not have any generally agreed-on restrictions. However, many applications of Markov chains employ finite or countably infinite state spaces, besides time-index and state-space parameters, there are many other variations and generalizations. For simplicity, most of this article concentrates on the discrete-time, discrete state-space case, the changes of state of the system are called transitions.
The probabilities associated with state changes are called transition probabilities. The process is characterized by a space, a transition matrix describing the probabilities of particular transitions. By convention, we assume all possible states and transitions have been included in the definition of the process, so there is always a next state, and the process does not terminate
Lean manufacturing or lean production, often simply lean, is a systematic method for the elimination of waste within a manufacturing system. Lean takes into account waste created through overburden and waste created through unevenness in work loads, working from the perspective of the client who consumes a product or service, value is any action or process that a customer would be willing to pay for. Lean manufacturing makes obvious what adds value, by reducing everything else and this management philosophy is derived mostly from the Toyota Production System and identified as lean only in the 1990s. TPS is renowned for its focus on reduction of the original Toyota seven wastes to improve overall customer value, the steady growth of Toyota, from a small company to the worlds largest automaker, has focused attention on how it has achieved this success. Lean principles are derived from the Japanese manufacturing industry, the term was first coined by John Krafcik in his 1988 article, Triumph of the Lean Production System, based on his masters thesis at the MIT Sloan School of Management.
Krafcik had been a quality engineer in the Toyota-GM NUMMI joint venture in California before joining MIT for MBA studies, a complete historical account of the IMVP and how the term lean was coined is given by Holweg. For many, lean is the set of tools that assist in the identification, as waste is eliminated quality improves while production time and cost are reduced. Techniques to improve flow include production leveling, pull production and the Heijunka box, the difference between these two approaches is not the goal itself, but rather the prime approach to achieving it. The implementation of smooth flow exposes quality problems that already existed, the advantage claimed for this approach is that it naturally takes a system-wide perspective, whereas a waste focus sometimes wrongly assumes this perspective. Both lean and TPS can be seen as a connected set of potentially competing principles whose goal is cost reduction by the elimination of waste. Thus what one sees today is the result of a need driven learning to improve where each step has built on previous ideas, from this perspective, the tools are workarounds adapted to different situations, which explains any apparent incoherence of the principles above.
Also known as the mass production, the TPS has two pillar concepts, Just-in-time or flow, and autonomation. Adherents of the Toyota approach would say that the smooth flowing delivery of value achieves all the improvements as side-effects. The other of the two TPS pillars is the human aspect of autonomation, whereby automation is achieved with a human touch. In this instance, the human touch means to automate so that the machines/systems are designed to aid humans in focusing on what the humans do best. These concepts of flexibility and change are principally required to allow production leveling, using tools like SMED, the flexibility and ability to change are within bounds and not open-ended, and therefore often not expensive capability requirements. More importantly, all of concepts have to be understood, appreciated. The cultural and managerial aspects of lean are possibly more important than the tools or methodologies of production itself
A system is a set of interacting or interdependent component parts forming a complex or intricate whole. Every system is delineated by its spatial and temporal boundaries and influenced by its environment, described by its structure and purpose and expressed in its functioning. Alternatively, and usually in the context of social systems. The term system comes from the Latin word systēma, in turn from Greek σύστημα systēma, whole compounded of several parts or members, according to Marshall McLuhan, System means something to look at. You must have a high visual gradient to have systematization. In philosophy, prior to Descartes, there was no system, in the 19th century the French physicist Nicolas Léonard Sadi Carnot, who studied thermodynamics, pioneered the development of the concept of a system in the natural sciences. In 1824 he studied the system which he called the substance in steam engines. The working substance could be put in contact with either a boiler, in 1850, the German physicist Rudolf Clausius generalized this picture to include the concept of the surroundings and began to use the term working body when referring to the system.
The biologist Ludwig von Bertalanffy became one of the pioneers of the systems theory. Norbert Wiener and Ross Ashby, who pioneered the use of mathematics to study systems, in the 1980s John H. Holland, Murray Gell-Mann and others coined the term complex adaptive system at the interdisciplinary Santa Fe Institute. Environment and boundaries Systems theory views the world as a system of interconnected parts. One scopes a system by defining its boundary, this means choosing which entities are inside the system, one can make simplified representations of the system in order to understand it and to predict or impact its future behavior. These models may define the structure and behavior of the system and human-made systems There are natural and human-made systems. Natural systems may not have an apparent objective but their behavior can be interpreted as purposefull by an observer, human-made systems are made to satisfy an identified and stated need with purposes that are achieved by the delivery of wanted outputs.
Their parts must be related, they must be designed to work as a coherent entity – otherwise they would be two or more distinct systems, Theoretical framework An open system exchanges matter and energy with its surroundings. Most systems are open systems, like a car, a coffeemaker, a closed system exchanges energy, but not matter, with its environment, like Earth or the project Biosphere2 or 3. An isolated system exchanges neither matter nor energy with its environment, a theoretical example of such system is the Universe. Inputs are consumed, outputs are produced, the concept of input and output here is very broad
In probability and statistics, a random variable, random quantity, aleatory variable, or stochastic variable is a variable quantity whose value depends on possible outcomes. It is common that these outcomes depend on physical variables that are not well understood. For example, when you toss a coin, the outcome of heads or tails depends on the uncertain physics. Which outcome will be observed is not certain, of course the coin could get caught in a crack in the floor, but such a possibility is excluded from consideration. The domain of a variable is the set of possible outcomes. In the case of the coin, there are two possible outcomes, namely heads or tails. Since one of these outcomes must occur, thus either the event that the coin lands heads or the event that the coin lands tails must have non-zero probability, a random variable is defined as a function that maps outcomes to numerical quantities, typically real numbers. In this sense, it is a procedure for assigning a numerical quantity to each outcome, contrary to its name.
What is random is the physics that describes how the coin lands. A random variables possible values might represent the possible outcomes of a yet-to-be-performed experiment and they may conceptually represent either the results of an objectively random process or the subjective randomness that results from incomplete knowledge of a quantity. The mathematics works the same regardless of the interpretation in use. A random variable has a probability distribution, which specifies the probability that its value falls in any given interval, two random variables with the same probability distribution can still differ in terms of their associations with, or independence from, other random variables. The realizations of a variable, that is, the results of randomly choosing values according to the variables probability distribution function, are called random variates. The formal mathematical treatment of random variables is a topic in probability theory, in that context, a random variable is understood as a function defined on a sample space whose outputs are numerical values. A random variable X, Ω → E is a function from a set of possible outcomes Ω to a measurable space E.
The technical axiomatic definition requires Ω to be a probability space, a random variable does not return a probability. The probability of a set of outcomes is given by the probability measure P with which Ω is equipped. Rather, X returns a numerical quantity of outcomes in Ω — e. g. the number of heads in a collection of coin flips
Industrial engineering is a branch of engineering which deals with the optimization of complex processes, systems or organizations. Industrial engineers work to eliminate waste of time, materials, man-hours, machine time, according to the Institute of Industrial and Systems Engineers, they figure out how to do things better, they engineer processes and systems that improve quality and productivity. Some engineering universities and educational agencies around the world have changed the term industrial to broader terms such as production or systems, the various topics concerning industrial engineers include, Process engineering, operation and optimization of chemical and biological processes. Systems engineering, a field of engineering that focuses on how to design. Safety engineering, a discipline which assures that engineered systems provide acceptable levels of safety. Value engineering, a method to improve the value of goods or products. Quality engineering, a way of preventing mistakes or defects in manufactured products, Project management, is the process and activity of planning, organizing and controlling resources and protocols to achieve specific goals in scientific or daily problems.
It includes the movement and storage of raw materials, work-in-process inventory, the practice of designing products, systems or processes to take proper account of the interaction between them and the people that use them. Logistics, the management of the flow of goods between the point of origin and the point of consumption in order to some requirements. Many of the tools and principles of engineering can be applied to the configuration of work activities within a project. The application of engineering and operations management concepts and techniques to the execution of projects has been thus referred to as Project Production Management. Traditionally, an aspect of industrial engineering was planning the layouts of factories and designing assembly lines. And now, in manufacturing systems, industrial engineers work to eliminate wastes of time, materials, energy. There is a consensus among a historian that the roots of the Industrial Engineering Profession date back to the Industrial Revolution.
The concept of the system had its genesis in the factories created by these innovations. Eli Whitney and Simeon North proved the feasibility of the notion of Interchangeable parts in the manufacture of muskets, under this system, individual parts were mass-produced to tolerances to enable their use in any finished product. The result was a significant reduction in the need for skill from specialized workers, frederick Taylor is generally credited as being the father of the Industrial Engineering discipline. He earned a degree in engineering from Stevens University