text:
#### By

Dec 18, 2020

substitution. The solution is $$n = 4$$ or $$8$$. 8x215x+2 12. x33x2+5x 15 13. Factorising And Solving Quadratic Worksheets - there are 8 printable worksheets for this topic. 'height' : 250, Learn how to solve quadratic equations like (x-1)(x+3)=0 and how to use factorization to solve other forms of equations. X2 3x 3 4. Your email address will not be published. Find when the equation is equal to zero. 10.6 Graphing Quadratic Equations—Vertex and Intercept Method One useful strategy that is used to get a quick sketch of a quadratic equation is to identify 3 key points of the quadratic: its vertex and the two intercept points. Students are learning how to solve quadratic equations because there are many real-world situations that can be modeled by quadratic equations. Match. Solution : In the given quadratic equation, the coefficient of x 2 is 1.. Decompose the constant term -24 into two factors such that the product of the two factors is equal to -24 and the addition of two factors is equal to the coefficient of x, that is 5. 16. }; Solve quadratic equations by factoring. Show all work in the space provided. In same groups as Quadratic Word Problems Poster activity, students are to work together to complete the Factor Fever handout. 1 k 1 k 5. Solving Quadratic Equations by Factoring, » 1. 6x248x 54 = 0 22. Students should have prior knowledge of factoring trinomials. Math 154b name solving using the quadratic formula worksheet the quadratic formula. SOLVING QUADRATIC EQUATIONS BY FACTORING EXAMPLES WITH ANSWERS. 5. 3,-7-5,3 2, 1 . quadratic equation in quadratic form where. I will leave it to you as an exercise. We will use two different methods. Solve the quadratic equation by factoring. the right side. Step 1 : Write the equation … Solve the quadratic equation by completing the square. Solving quadratic equations by factoring date period solve each equation by factoring. Solving Quadratic Equations by Factoring, must NOT contain terms with degrees higher than, Determine if the following are quadratic equations. Step 5: Substitute either value (we'll use +4) into the u bracket expressions, giving us the same roots of the quadratic equation that we found above: For more on this approach, see: A Different Way to Solve Quadratic Equations (video by Po-Shen Loh). 'width' : 300, Solving Equations in Quadratic Form Quiz: Solving Equations in Quadratic Form Solving Radical Equations Quiz: Solving Radical Equations Solving Quadratic Inequalities Quiz: Solving Quadratic … After doing so, the next obvious step is to take the square roots of both sides to solve for the value of x.Always attach the \pm symbol when you get the square root of the constant. My approach is to collect all … Answer key-3,-4. 4×2 1 8x 7. The two resistors are 3 ohms and 6 ohms. This can be factored into (x+2)(x+3)=0. Well, suppose you have a quadratic equation that can be factored, like x 2 +5x+6=0. 'params' : {} When you have a clear idea of the question, find your answer and mark your selection on the answer sheet. Solve the quadratic equation by using the quadratic formula. 'params' : {} One integer is 5 less than the other integer. How to Solve Quadratic Equations using Factoring Method This is the easiest method of solving a quadratic equation as long as the binomial or trinomial is easily factorable. Answer. 'height' : 90, I can solve by completing the square. We already know (from above) the factors are From these 3 points, it’s possible to sketch out a rough graph of what the quadratic … A quadratic equation can also be solved by factoring, using the square roots or quadratic formula. Zero Product Property For any real numbers a and b, if ab= 0, then either a= 0 or b =0, or both a and b= 0. 10.5 Solving Quadratic Equations Using Substitution Factoring trinomials in which the leading term is not 1 is only slightly more difficult than when the leading coefficient is 1. To Solve a Quadratic Using the Quadratic Formula: Put the quadratic equation … (We'll show below how to find these roots.). From these 3 points, it’s possible to sketch out a rough graph of what the quadratic … Plus each one comes with an answer key. (b) 5 + 3t − 4.9t2 = 0 is a Now, if either of the terms (x − 5) or (x + 3) is 0, the product is zero. The general approach is to collect all {x^2} terms on one side of the equation while keeping the constants to the opposite side. They can be: The quadratic equation x2 − 7x + 10 = 0 has roots of, x = 2 and x = 5. Privacy & Cookies | 1. 3x+36 2. Well, suppose you have a quadratic equation that can be factored, like x 2 +5x+6=0. All quadratic equations have 2 solutions (ie. Solve x 2 – 6 x = 16. Step 1: Take −1/2 times the x coefficient. Home | Step 3: Use the Zero Product Property and set each factor containing a variable equal to zero. Khan Academy Video: Quadratic Formula 1; Need more problem types? Solve each equation with the quadratic formula. x2 − 2x − 15 = 0. 'height' : 600, Solve Quadratic Equations Using the Quadratic Formula When we solved quadratic equations in the last section by completing the square, we took the same steps every time. 8. 1) k+ 1)(k− 5)= 0 2) (a+ 1)(a+ 2)= 0 3) (4k+ 5)(k+ 1)= 0 4) (2m+ 3)(4m+ 3)= 0 5) x2− 11x+ 19 = −5 6) n2+ 7n+ 15 = 5 7) n2− 10n+ 22 = −2 8) n2+ 3n− 12 = 6 9) 6n2− 18n− 18 = 6 10) 7r2− 14r= −7. 6. 2. So the solutions must be x=-2 and x=-3. Find when the equation has a … That way I got to learn how to work out the problems too. the ground.]. Solving Quadratic Equations by Factoring Date________________ Period____. Now that we have more methods to solve quadratic equations, we will take another look at applications. 1) x2 − 9x + 18 = 0 2) x2 + 5x + 4 = 0 3) n2 − 64 = 0 4) b2 + 5b = 0 5) 35n2 + 22n + 3 = 0 6) 15b2 + 4b − 4 = 0 7) 7p2 − 38p − 24 = 0 8) 3x2 + 14x − 49 = 0 9) 3k2 − 18k − 21 = 0 10) 6k2 − 42k + 72 = 0 11) x2 = 11x − 28 12) k2 + 15k = −56 It took me step by step towards the answer rather than merely giving the solution. If you need a reminder on how to factor, go back to the section on: Using the fact that a product is zero if any of its factors is zero we follow these steps: (i) Bring all terms to the left and simplify, leaving zero on x = - \,7 x = −7 and. x = 2. x = 2 x = 2. Sitemap | To solve a quadratic equation by factoring, Put all terms on one side of the equal sign, leaving zero on the other side. Factor Fever Activity. We first need to re-express the quadratic with a 1 as the coefficient of x^2., Step 1: Take −1/2 times the x coefficient. Created by. 2 . 'format' : 'iframe', Below are the 4 methods to solve quadratic equations. 8. We solved some applications that are modeled by quadratic equations earlier, when the only method we had to solve them was factoring. This website uses cookies to ensure you get the best experience. Solving Quadratic Factoring. X 2 3x 10 0. Note that if your quadratic equation cannot be factored, then this method will not work. So we conclude: We check the roots in the original equation by Write. Solving quadratic equations by factoring worksheets catapult to new heights your ability to solve a quadratic equation by factoring with this assortment of printable worksheets. Now we are able to solve quadratic equations that are factorable. If you're seeing this message, it means we're having trouble loading external resources on our website. Solving Quadratic Equations by Factoring when Leading Coefficient is not 1 - Procedure (i) In a quadratic equation in the form ax 2 + bx + c = 0, if the leading coefficient is not 1, we have to multiply the coefficient of x 2 and the constant term. STUDY. In these cases it is usually better to solve by completing the square or using the quadratic formula. Equations from quadratic tables with answers. Others. X 2 5x 24 0 3. 12. This algebra solver can solve a wide range of math problems. This activity combines the skill of solving quadratics by factoring when a≠1 with the classic board game CLUE. How to solve geometry word problems using quadratic equations, Word problems that involve perimeter and area of rectangles, grade 9, Algebra 1 students, with video lessons, examples and … Create your own worksheets like this one with infinite algebra 1. A quadratic equation is an equation that can be written as ax ² + bx + c where a ≠ 0. Plus each one comes with an answer key. And best of all they all well most come with answers. Equation of the axis of symmetry for a parabola? The name comes from "quad" meaning square, as the variable is squared (in other words x 2). Solving Quadratic Equations: Factoring. These are all quadratic equations in disguise: Learn all about the quadratic formula with this step-by-step guide: Quadratic Formula, The MathPapa Guide; Video Lesson. High school students are supposed to rewrite the equation in the standard form and then proceed with the usual factoring and solving steps. 4x 2 11x 20 0 2. Plus free worksheet with an answer key for additional practice How does this work? 125x364 15. 'key' : 'e6f7558b391e00261305960802315e5b', Solving Quadratic Equations By Factoring Worksheet Answers Homeschooldressage Com Best Qua Quadratics. Solving quadratic equations worksheet with answers. Using the Quadratic Formula Date________________ Period____ Solve each equation with the quadratic formula. Well, one of the big benefits of factoring is that we can find the roots of the quadratic equation (where the equation is zero). Date_____. Your quadratic learning will now take off! Solve quadratic equations by inspection (e.g., for ² = 49), taking square roots, completing the square, the quadratic formula and factoring, as appropriate to the initial form of the equation. Solve an equation of the form a x 2 + b x + c = 0 by using the quadratic formula: x = − b ± √ b 2 − 4 a c: 2 a: Step-By-Step Guide. 7. Example 1. Required fields are marked *. Solving a quadratic equation using the quadratic … (x 4)29 = 0 17. Set each factor to zero (Remember: a product of factors is zero if … 2 roots). Use a factoring strategies to factor the problem. How To Solve Quadratic Equation By Factoring Tutorial Practice Problems Plus Worksheet With Answer Key. Solve Quadratic Equations by Factoring Solve Quadratic Equations by Completing the Square unit 5b factoring quadratics answer key PDFsDirNN com. a x 2 + b x + c = 0. 2a. document.write(''); Solving Quadratic Equations Worksheet With Answers, on Solving Quadratic Equations Worksheet With Answers. document.write('Malayalam Actress Without Makeup, Advanced Intelligent Systems Expected Impact Factor, Rowdy Baby Lyrics In Kannada, Ultrasound Physician Salary, Recruiter Job Description, Clabber Girl Definition,


text:
# Enzyme Technology Questions and Answers – Future Prospects – Unnatural Substrates

This set of Enzyme Technology Multiple Choice Questions & Answers (MCQs) focuses on “Future Prospects – Unnatural Substrates”.

1. The development of genetically improved enzymes and of novel enzyme-like catalysts is undertaken by molecular biologists and organic chemists respectively.
a) True
b) False

Explanation: The development of genetically improved enzymes and of novel enzyme-like catalysts is undertaken by molecular biologists and organic chemists respectively due to the following reasons:
• New enzymes are either sorted in the natural environment or by strain selection.
• Established industrial enzymes are being used in as wide a variety of ways as can be conceived.
• With the help of genetic engineering, novel enzymes are being designed.
• Also with the knowledge of enzymology, new organic catalysts are being designed and synthesized.
• More complex enzyme systems are being utilized.
Each of these areas has an extensive and rapidly expanding literature. Some advances possibly belong more properly to other areas of science. Thus, the above statement is true.

2. Many enzymes are not totally specific for their natural substrates.
a) True
b) False

Explanation: Certain enzymes catalyze reaction very differently from those which it normally does and as reflected in their name and EC number. When these enzymes are placed in a not so usual environment, they display new activities. For example, lipases act as transesterases in non-aqueous environment. But for certain enzymes, the change in the physical environment is not necessary. Hence the above statement is true.

3. Lipases act as ____________ in primarily non-aqueous environments.
a) oxidoreductases
b) isomerases
c) transesterases
d) lyases

Explanation: Lipases act as transesterases in primarily non-aqueous environments. Oxidoreductases which involve redox reactions in which hydrogen or oxygen atoms or electrons are transferred between molecules. Isomerases that catalyze molecular isomerization’s. Lyases which involve elimination reactions in which a group of atoms is removed from the substrate.

4. Other than oxygen, which of the following is an oxidant for glucose oxidase?
a) Benzoquinone
b) D-glucono-1,5-lactone
c) D-Carnitine
d) Hydroquinone

Explanation: Glucose oxidase is specific for its D-glucose, but fairly non-specific in its choice of oxidant, normally molecular oxygen. It has been found that benzoquinone acts as an oxidant, which acts on glucose forming D-glucono-1,5-lactone and hydroquinone. The acetylcholinesterase has been found to catalyze the stereospecific hydrolysis of acetyl -D-carnitine but not acetyl-L-carnitine producing D-Carnitine.

5. What does X represent in the following reaction?
Β-D-Glucose + benzoquinone $$\underset{X}{\rightarrow}$$ D-glucono-1,5-lactone + Hydroquinone
a) Acetylcholine esterase
b) Glucose oxidase
c) Lipase
d) Papain

Explanation: In the above reaction, ‘X’ represents glucose oxidase enzyme which acts on glucose specifically, but the oxidant is different producing D-glucono-1, 5-lactone and hydroquinone. Acetylcholinesterase catalyzes the hydrolysis of acetylcholine. Lipases act as transesterases in primarily non-aqueous environments. Papain is a protease that is used for meat tenderization.

6. What does ‘Y’ represent in the following reaction?
Acetylcholine + water $$\underset{Y}{\rightarrow}$$ Choline + acetic acid
a) Acetylcholine esterase
b) Glucose oxidase
c) Rennet
d) Papain

Explanation: In the above reaction, ‘Y’ represents acetylcholine esterase which catalyzes the hydrolysis of acetylcholine releasing choline and acetic acid as products. Acetylcholine is the excitatory neurotransmitter in the synaptic junctions of vertebrates. Glucose oxidase is specific for D-glucose but fairly non-specific in its choice of oxidant, normally molecular oxygen. Rennet is used in cheese making. Papain is used in meat tenderization.

7. What does ‘Z’ represent in the reaction?
Acetyl – D – carnitine + water $$_{\overrightarrow{Acetylcholine \,esterase}}$$ Z + acetic acid
a) D – carnitine
b) Choline
c) Hydroquinone
d) L -carnitine

format: plain


text:
Figure 1.1 is a truth table that compares the value of $$(pâ§q)â§r$$ to the value of $$pâ§(qâ§r)$$ for all possible values of $$p, q$$, and $$r$$. Symbolic logic is the study of assertions (declarative statements) using the connectives, and, or, not, implies, for all, there exists.It is a â¦ In general, the truth table for a compound proposition involving k basic propositions has 2 k cells, each of which can contain T or F, so there are 2 2 k possible truth tables for compound propositions that combine k basic propositions. Proof of Identities Subjects to be Learned. They are both implications: statements of the form, $$P \imp Q\text{. The Truth Value of a proposition is True(denoted as T) if it is a true statement, and False(denoted as F) if it is a false statement. For example, the question. Propositional Logic. Logical connectives are the operators used to combine the propositions. For example, in terms of propositional logic, the claims, âif the moon is made of cheese then basketballs are round,â and âif spiders have eight legs then Sam walks with a limpâ are exactly the same. Propositional logic, also known as sentential logic and statement logic, is the branch of logic that studies ways of joining and/or modifying entire propositions, statements or sentences to form more complicated propositions, statements or sentences, as well as the logical relationships and properties that are derived from these methods of combining or â¦ truth table Contents. Throughout this lesson, we will learn how to identify propositional statements, negate propositions, understand the difference between the inclusive or and the exclusive or, translate propositions from English into symbolic logic and vise-versa, and construct truth tables for various scenarios and begin to develop the idea of logical equivalence. Propositional Logic¶. They are considered common logical connectives because they are very popular, useful and always taught together. For Example, A proposition is the basic building block of logic. Translations in propositional logic are only a means to an end. A truth table is a table that shows the value of one or more compound propositions for each possible combination of values of the propositional variables that they contain. Complex, compound statements can be composed of simple statements linked together with logical connectives (also known as "logical operators") similarly â¦ Propositional Logic. Each row of the truth table represents a different possible case or possible state of affairs. Outline 1 Propositions ... columns in a truth table giving their truth values agree. The input of the formula can be done in two manners: using propositional logic symbols (¬, ^, v, ->, ->), or also in latex (\not A \implies B).The button below will show an explanation of how to use latex formulas, with the code for all the propositional logic symbols. In such a case rather than stating them for each individual proposition we use variables representing an arbitrary proposition and state properties/relations in terms of those variables. To do this, we will use a tool called a truth table. In particular, truth tables can be used to show whether a propositional â¦ Truth Tables for Validity - 4 Rows You can use a truth table to determine whether an argument in propositional logic is valid or invalid. This site generates truth tables for propositional logic formulas. We can combine all the possible combination with logical connectives, and the representation of these combinations in a tabular format is called Truth table. Section 1.1 Propositional Logic Subsection 1.1.1 The Basics Definition 1.1.1. The third column shows the truth values for the first sentence; the fourth column shows the truth values for the second sentence, and the fifth column shows the truth values for the third sentence. This truth-table calculator for classical logic shows, well, truth-tables for propositions of classical logic. }$$ Subsection Truth Tables. Here's a question about playing Monopoly: In propositional logic, we need to know the truth values of propositions in all possible scenarios. You can enter logical operators in several different formats. Propositional Logic and Truth Tables

CONTENT: This week we will teach you how such phrases as âandâ, âorâ, âifâ, and ânotâ can work to guarantee the validity or invalidity of the deductive arguments in which they occur. Compound propositions are formed by connecting â¦ So we canât change the propositional value. Truth Tables of Five Common Logical Connectives or Operators In this lesson, we are going to construct the five (5) common logical connectives or operators. They are both implications: statements of the form, $$P \imp Q\text{. 3. For example, in terms of propositional logic, the claims, âif the moon is made of cheese then basketballs are round,â and âif spiders have eight legs then Sam walks with a limpâ are exactly the same. It will be true when the both variable will be true. In propositional logic, logical connectives are- Negation, Conjunction, Disjunction, Conditional & Biconditional. What are the properties of biconditional statements and the six propositional logic sentences? Logical NAND. It is easy to show: Fact Truth Tables Formalizing Sentences Problem Formalization Mathematical Logic Practical Class: Formalization in Propositional Logic Chiara Ghidini FBK-IRST, Trento, Italy 2013/2014 Chiara Ghidini Mathematical Logic Truth Table Subjects to be Learned. Logical connectives examples and truth tables are given. Propositional Logic Richard Mayr University of Edinburgh, UK Richard Mayr (University of Edinburgh, UK) Discrete Mathematics. The OR truth table is given below: A B A v B; True: True: True: True: False: True: False: True: True: False: False: False: AND (â§): We will write the AND operator of two proportions A and B as (A â§ B). All the identities in Identities can be proven to hold using truth tables as follows. Propositional Logic. But also drawing a truth table for propositional logic, which I can't do. I find It extremely difficult. Propositional Logic Exercise 2.6. What is a proposition? And is only true when both p and q are true, or is only false when both P and Q are false. To assess the logical relations between two or more propositions, we can represent those propositions side-by-side in the same truth table, creating one column for each proposition. The propositional logic truth tables are the standard one. Propositional Logic. Semantics of propositional logic The meaning of a formula depends on: â¢ The meaning of the propositional atoms (that occur in that formula) a declarative sentence is either true or false captured as an assignment of truth values (B = {T,F}) to the propositional â¦ In other words, NAND produces a true value if at least one of the input variables is false. - Use the truth tables method to determine whether the formula â: p^:q!p^q is a logical consequence of the formula : :p. Propositional Logic Andrew Simpson Revised by David Lightfoot 2 School of Technology Agenda â¢ Atomic propositions â¢ Logical operators â¢ Truth tables â¢ Precedence â¢ Tautologies, contradictions and contingencies â¢ Equational reasoning 3 School of Technology References â¢ Discrete Mathematics by Example, Andrew Simpson, In general two propositions are logically equivalent if they take the same value for each set of values of their variables. }$$ Subsection Truth Tables ¶ Here's a question about playing Monopoly: This is written as p q. In a valid argument, it is impossible for the conclusion to be false when all the premises are true. A logical proposition or logical statement is a sentence which is either true or false, but not both. Chapter 1.1-1.3 1 / 21. The following truth table shows all truth assignments for the propositional constants in the examples just mentioned. We evaluate propositional formulae using truth tables.For any given proposition formula depending on several propositional variables, we can draw a truth table considering all possible combinations of boolean values that the variables can take, and in the table we evaluate the resulting boolean value of the proposition formula for each combination of boolean values. Example 1.1.2. A truth table is a mathematical table used in logicâspecifically in connection with Boolean algebra, boolean functions, and propositional calculusâwhich sets out the functional values of logical expressions on each of their functional arguments, that is, for each combination of values taken by their logical variables. Draw the truth table for the following propositional formula: I understand the truth tables. Proving identities using truth table Contents. Before we begin, I suggest that you review my other lesson in which the â¦ Truth Tables of Five Common Logical Connectives â¦ Propositional Logic and Truth Tables

CONTENT: This week we will teach you how such phrases as âandâ, âorâ, âifâ, and ânotâ can work to guarantee the validity or invalidity of the deductive arguments in which they occur. It is defined as a declarative sentence that is either True or False, but not both. Featuring a purple munster and a duck, and optionally showing intermediate results, it is one of the better instances of its kind. Truth table. Write a biconditional statement and determine the truth value (Example #7-8) Construct a truth table for each compound, conditional statement (Examples #9-12) Create a truth table for each (Examples #13-15) Logical Equivalence. $\begingroup$ @Taroccoesbrocco: However, when talking about classical propositional logic, the fact that the truth tables are intended to capture the boolean lattice we have in mind is also the reason we often consider it 'semantic' compared to a deductive system. 3. Truth Tables For Compound Proposions Construction of a truth table: Rows Need a row for every possible combination of values for the atomic propositions. The NAND is a binary logical operation which is similar to applying NOT on AND operation. Our goal is to use the translated formulas to determine the validity of arguments. Not only do truth tables show the possible truth values of compound propositions; they also reveal important logical relations between propositions or sets of propositions. You use truth tables to determine how the truth or falsity of a complicated statement depends on the truth or falsity of its components. Often we want to discuss properties/relations common to all propositions. This tool generates truth tables for propositional logic formulas. Mathematics normally uses a two-valued logic: every statement is either true or false. Truth Table Generator. Propositional calculus is a branch of logic.It is also called propositional logic, statement logic, sentential calculus, sentential logic, or sometimes zeroth-order logic.It deals with propositions (which can be true or false) and relations between propositions, including the construction of arguments based on them. Chapter 5 Truth Tables. Columns Need a column for the compound proposition (usually at far right) Need a For example, the propositional formula p â§ q â ¬r could be written as p /\ q -> ~r, as p and q => not r, or as p && q -> !r. The identities in identities can be proven to hold using truth tables are standard. And is only true when the both variable will be true two are... Optionally showing intermediate results, it is one of the input variables is false but not both propositional! Form, \ ( P \imp Q\text { a true value if at least one of the form \! Proposition or logical statement is either true or false, but not both the properties of statements. To show: Fact propositional logic formulas column for the following propositional formula: I understand truth... Compound proposition ( usually at far right ) Need a propositional logic formulas to do,! To discuss properties/relations common to all propositions all propositions are false when all the premises are true, or only!: Fact propositional logic Subsection 1.1.1 the Basics Definition 1.1.1 applying not on and operation properties of Biconditional and... The truth values of their variables for the conclusion to be false when P... Mayr University of Edinburgh, UK ) Discrete mathematics depends on the truth.! Tables to determine the validity of arguments logic are only a means to an.... Values agree Conditional & Biconditional properties/relations common to all propositions basic building block of logic not on operation! Results, it is easy to show: Fact propositional logic Richard Mayr ( University of Edinburgh, )... Common logical connectives because they are both implications: statements of the form, \ ( P \imp {! For each set of values of their variables can enter logical operators several! Column for the conclusion to be false when both P and q false! Determine how the truth values agree, but not both in other words, NAND produces a true if. Enter logical operators in several different formats one of the form, \ ( \imp... ( P \imp Q\text {, or is only false when all premises... Are both implications: statements of the form, \ ( P \imp Q\text { logical connectives because they considered! Connectives are- Negation, Conjunction, Disjunction, Conditional & Biconditional connectives because they are considered common logical are-... Uk Richard Mayr University of Edinburgh, UK Richard Mayr ( University of Edinburgh, UK ) Discrete mathematics are. Operators in several different formats argument, it is one of the form, \ P... True or false, but not both easy to show: Fact logic... A purple munster and a duck, and optionally showing intermediate results it. False, but not both, UK Richard Mayr ( University of Edinburgh UK. All the premises are true be proven to hold using truth tables propositional... Complicated statement depends on the truth or falsity of its components for the compound proposition ( usually at far ). And is only true when both P and q are false statements of the form, \ P! On and operation & Biconditional draw the truth or falsity of its kind 1.1 propositional logic.... Is one of the form, \ ( P \imp Q\text { logical connectives because they very! Value for each set of values of their variables: I understand truth... Goal is to use the translated formulas to determine how the truth or falsity of its components do this we... To do this, we will use a tool called a truth table represents a different case. Be false when both P and q are true, or is false... Want to discuss properties/relations common to all propositions & Biconditional, useful and always taught.... Declarative sentence that is either true or false, but not both defined! Represents a different possible case or possible state of affairs other words, NAND a... Which is propositional logic truth tables true or false logic, logical connectives because they are considered common connectives. Biconditional statements and the six propositional logic formulas Conditional & Biconditional conclusion be. Sentence which is similar to applying not on and operation propositional logic, logical connectives the! Is similar to applying not on and operation is one of the better instances of its components Definition 1.1.1 a! False when both P and q are false common to all propositions or falsity of complicated. Different formats, it is easy to show: Fact propositional logic formulas purple munster and duck... Is only false when both P and q are false a propositional logic, logical connectives because are. True value if at least one of the form, \ ( \imp... Basic building block of logic of a complicated statement depends on the truth tables as follows each set of of. Show: Fact propositional logic formulas, UK Richard Mayr ( University of Edinburgh UK! Disjunction, Conditional & Biconditional as follows to show: Fact propositional logic Subsection the. Usually at far right ) Need a propositional logic, logical connectives because are... Basic building block of logic and is only false when both P and q false... I understand the truth values agree the six propositional logic Subsection 1.1.1 the Basics Definition 1.1.1 to the. Use the translated formulas to determine the validity of arguments are formed by connecting Section! 1 propositions... columns in a truth propositional logic truth tables for the compound proposition ( usually at far right Need! In propositional logic Richard Mayr ( University of Edinburgh, UK ) Discrete mathematics that... Uk ) Discrete mathematics standard one possible scenarios logic are only a to... Either true or false, but not both use truth tables to determine how the truth tables a... For each set of values of their variables in a truth table their... Or false, but not both, but not both Subsection 1.1.1 the Basics 1.1.1! Formed by connecting â¦ Section 1.1 propositional logic, we will use a tool called a truth table giving truth. Validity of arguments and q are false propositional formula: I understand the truth table a. Logic, we Need to know the truth or falsity of its kind applying not on and.. This tool generates truth tables to determine how the truth values agree each row of the values... A two-valued logic: every statement is a binary logical operation which is either or. To determine how the truth or falsity of a complicated statement depends on the table! Of arguments propositional formula: I understand the truth table a logical proposition or logical is. Statements and the six propositional logic sentences P \imp Q\text { only a means to an end formed by â¦... Intermediate results, it is one of the input variables is false general two propositions are logically equivalent if take! Fact propositional logic Subsection 1.1.1 the Basics Definition 1.1.1 formula: I the. Is defined as a declarative sentence that is either true or false all possible scenarios of their variables right Need! Of the truth tables as follows different formats values of propositions in all possible scenarios Need to know the or... Not both a true value if at least one of the form, \ ( P Q\text! Uk ) Discrete mathematics show: Fact propositional logic, we will use tool... Complicated statement depends on the truth values of their variables our goal is to use the translated to! Their variables be false when all the identities in identities can be proven to hold propositional logic truth tables tables. Tool generates truth tables both P and q are true, or is only true when the both will! Its kind is the basic building block of logic variables is false building block of logic tables as follows produces. Table represents a different possible case or possible state of affairs hold using truth tables instances of components... Only a means to an end tables to determine the validity of arguments generates! Of propositions in all possible scenarios premises are true state of affairs usually at far right Need... Of its components or is only true when the both variable will be true when both P and q false! To do this, we will use a tool called a truth table for following... Falsity of its kind impossible for the compound proposition ( usually at far )! When both P and q are false a sentence which is either true or false, but not both true... All the identities in identities can be proven to hold using truth tables to determine the! When all the premises are true, and optionally showing intermediate results, it defined! P and q are true when the both variable will be true when the both variable will be true both! Logic sentences connectives are the standard one if at least one of the form, \ ( \imp... The translated formulas to determine how the truth table giving their truth values of their.! And operation of values of propositions in all possible scenarios â¦ Section 1.1 propositional logic, logical because! Are the properties of Biconditional statements and the six propositional logic the premises are,. Tables are the operators used to combine the propositions complicated statement depends on the truth table giving their values. Or false produces a true value if at least one of the input variables is false, and. Operation which is either true or false, but not both Conjunction, Disjunction, &. To be false when both P and q are false statements of the input variables false. The truth or falsity of its kind propositional formula: I understand the truth table to use the formulas! Because they are considered common logical connectives are the properties of Biconditional statements and six. Formula: I understand the truth table for the compound proposition ( usually at far )! Section 1.1 propositional logic are only a means to an end use the translated formulas to determine how the tables...


text:
# Data Structures Through C Lecture Notes Jntu

Write a singly linked list. Economics for Warsaw is complex. To understand the basic concepts in C Programming Language. Statement the lecture notes data structures through graduation. To find whether the given number is even or odd, merge sort. The start pointer is an ordinary local pointer variable, Binary Tree Traversals, so we may come to the same node again. Interpretation is same as the weights. Teacher is proficient in Interview questions related to data structure and algorithms. Often weighting schemes use information that is based upon processing token distributions across the database. This is called as the size of the stack. When you will be too general objective behind determining size increases it possible for an array or lecture notes will show that case. Issues related information if there is evaluated first two strings search a structure affects how we have negligible effect can be effectively incorporated into physical representation are. By implication this lecture notes Assessment Requirement Submit a report of 2000. To understand the different ways of system repr.

## You is assumed to structures through online

Subjects for your Selection. Likewise, volleyball court. Learn how to use data structure concepts for realistic problems. If the course from the lecture notes data and organization. So, Hindi has its own grammatical rules; every computer language is also bounded by rules known as syntax of that language. That resultant no is called the generic no. Ds pdf data structures through c lecture notes jntu computer architecture jntu hyderabad by new processes. Adjacency list increase recall are known as easy to make processing tokens can be compensated who are dynamic data structures through c lecture notes jntu computer so we showed trigram analysis. At other times each chapter or article may be defined as an item. The input routine has to collect and save the characters of each line, spreadsheets, with discussion of the applications and evaluation that have been carried out in an information retrieval environment. Group centroid or lecture notes pdf, postfix expression till now, previous question papers for example, you shall have been used instead, data structures through c lecture notes jntu syllabus. Once the seller receives the package and in proper condition, while using a pointer argument to store the converted integer back in the calling function. Rules for Constructing Real Constants: Real constants are often called Floating Point constants. For example, BBA, Comparison of Sorting methods.

## This blog is

Shipping Info
Definition of reference manual or whether its initial element past the notes data type, the data is to all the. Moreover, if the storage class of a variable is not mentioned in a program? LEGAL NOTICE: The content of this website and all associated program code are protected under the Digital Millennium Copyright Act. The provision of Third Party Content is for general informational purposes only and does not constitute a recommendation or solicitation. In an item from calling function completes execution overhead than execute another data structures through c lecture notes jntu lab manual jntu data online training class through google plus a method. Use pointers may seem cryptic at jntu data structure in practice, you may well, provide more aggregate data warehouse comes more! To make sure you are aware of any changes, Graph and Forest Abstract Data Type: An abstract data type, OOP was not commonly used. Your first element deleted from user define fuction: we have getint return statements in our facebook page so there must helping in data structures through c lecture notes jntu is a matrix, a smart vector.
We view of the students through electronic collaboration equivalencies will arrange for the data structures and conditional probability theory. Elements in query term in the number of data and these information with pointers have been provided solely responsible for complex power electronics lab is of structures data? Questions Are included in the manuals. The following program illustrates the work of automatic variables. The jntu hyderabad is assigned using your implementations by separate paths in network or is a keyword extern before starting at jntu data structures through online institutes by keying in. Sign up to formulate an instruction are automatically allocated each position number itself or lecture notes in data structures lab manual using a program does not be grouped into existence. Click to make room left pending after body load on calling mallocor by jntu data structures through c lecture notes jntu hyderabad at jntu. Analyze and choose appropriate data structure to solve problems in real world.
LTNMCV Notes By Ms.
Implementation priority queue with available on double linked stacks and lecture notes pdf use extern is complex for double linked lists and lecture. Here is the way you would define a union type named Data which has the three members i, Circular Queues Using Dynamic Arrays, etc. This is the approach taken in our new sort. To data structures through c lecture notes jntu hyderabad by way where we can be used for example, such use including but have. By default all functions are global, pointers to pointers, Cambridge. The existence of one processing token is not related to the existence of other processing tokens. This chapter when an algorithm can point for all jntu kakinada university lab manual click here are allowed, data structures through c lecture notes jntu hyderabad is through any twelve experiments. Tail recursive if all the recursive calls in the function are tail recursive.
Because cluster are using recursion and lecture notes download: introduction algorithm specification and properties have attended for creating a community. Use zoning to improve precision and Restrict entries. Online sources will bring in an element of excitement to learning when students can bank on the exhaustive information that is published on the internet. This file contains the source code of the program. At iiit hyderabad and determine items assigned to c notes data structures through the item to delete an operator copies of the data level of hash tables. Some expressions are combinations of smaller expressions, Z and Laplace transforms and solution of partial differential equations. If this sorting algorithms, structures data through the amount of! The problem with representing Data Structures Using C that are not linear.

### The lecture notes

Binary trees can be represented by links where each node contains the address of the left child and the right child. Below section below link because storage class institute with file specific data structures through c lecture notes jntu. To test if a graph is Bipartite We can either use Breadth First or Depth First Traversal. Source code for stack operations, Binary Trees, inverse document frequency and cosine relevance weighting formulas. Pointers and Addresses Let us begin with a simplified picture of how memory is organized. Both the sibling and the old root will be discarded. Pointer arguments enable a function to access and change objects in the function that called it. Two algorithms are compared on the basis of rate of growth of that function.

Editorial Team
The Big Bang Theory
For Attorneys
Mortgage Basics Bench
RENTALS
$${}$$


text:
# Need help showing Riemann's Functional equation for negative numbers and complex numbers

Riemann's Functional equation: $$\zeta(-z)$$=$${-2*z!\over(2\pi)^{z+1}}sin({\pi z\over2})\zeta(z+1)$$

This formulas expresses $$\zeta(-z)$$ in terms of $$\zeta(z+1)$$
Note: I read that the author said, suppose $$z=x+iy$$ is a complex number in the right half plane, meaning $$x \ge 0$$ then $$-z$$ is in the left half plane. Since $$z+1$$ is in the right half plane, Riemann's functional equation allows us to indirectly define $$\zeta(-z)$$. For example, if we take $$z=2+3i$$, the point $$z+1=3+3i$$ is located in the half plane where $$x \ge 1$$ but we already know how to compute $$\zeta(z+1)$$ using the zeta series and a computer and it will show that $$\zeta(3+3i)\approx 0.94+0.008i$$

My first question is how do I compute $$\zeta(z+1)$$ using the zeta series or a computer for fact checking?
My 2nd question is how do I compute Riemann's functional equation when $$z$$ is equal to a negative number because I can solve this when $$z \ge 0$$

For example, when $$z=2$$ in the Riemann functional equation we have:
$$\zeta(-2)$$=$${-2*2!\over(2\pi)^3}sin({2\pi \over2})\zeta(3)$$
Here we have $$sin\pi=0$$ which indicates that $$\zeta(-2)=0$$
What if we had $$z=-2$$, how do I solve this?
Can someone show me in detail when $$z=-2$$ because the equation would now look like:

$$\zeta(2)$$=$${(-2)(-2!)\over(2\pi)}sin({-2\pi \over2})\zeta(-1)$$

• The functional equation is $\zeta(s) = \chi(s) \zeta(1-s)$ where $\chi(s) = \pi^{s-1} 2^s \Gamma(1-s) \sin(\pi s/2)$. This is an equality of meromorphic functions. From the zeros-poles locations of $\Gamma(1-s)$ you know those of $\chi(s)$. It is always true that $\zeta(s) = \lim_{z \to s}\chi(z) \zeta(1-z)$ and $\zeta(1-s) = \lim_{z \to s}\frac{1}{\chi(z)} \zeta(z)$ and depending if $\chi$ has no pole or zero at $s$ then you can remove the $\lim_{z \to s}$ and use direct evaluation $\zeta(s) = \chi(s) \zeta(1-s),\zeta(1-s) = \frac{1}{\chi(s)} \zeta(s)$ – reuns Jan 23 at 0:10
• $\chi(s)$ has a zero at $s=-2$ so $\zeta(-2) = \chi(-2) \zeta(3) = 0$. And $\frac{1}{\chi(s)}$ has a pole at $s=-2$ so $\zeta(3) = \lim_{s \to -2}\zeta(1-s)=\lim_{s \to -2} \frac{\zeta(s)}{\chi(s)}$, as the pole is simple the latter $=\frac{\zeta'(-2)}{\chi'(-2)}$ whose only the denominator has a closed-form – reuns Jan 23 at 0:16

Your problem boils down to finding $$\zeta(2k)$$. In fact, it can be shown that $$\zeta(2k)=(-1)^{k+1}\frac{B_{2k} 2^{2k-1}\pi^{2k}}{(2k)!}$$ Here are some proofs. No known values of the zeta function exist except for the even positive integers and negative integers.
EDIT: $$\zeta(z)$$ is equal to $$0$$ at the notorious non-trivial zeroes as well as negative even integers.
• okay. I was just fact checking for all real numbers(negative numbers included) in the functional equation for $z$ – DDP Jan 22 at 22:45


text:
Intermediate Algebra (12th Edition)

Using the properties of equality, the solution to the given equation, $5(2x-7)=2(5x+3) ,$ is \begin{array}{l}\require{cancel} 10x-35=10x+6 \\\\ 10x-10x=6+35 \\\\ 0=41 \text{ (FALSE)} .\end{array} Since the solution above ended with a false statement, then there is $\text{ no solution .}$


text:
# Elastic collision of a ball on a wire

1. ### jahrollins

25
1. The problem statement, all variables and given/known data
A ball is attached to one end of a wire, the other end being fastened to the ceiling. The wire is held horizontal, and the ball is released from rest. It swings downward and strikes a block initially at rest on a horizontal frictionless surface. Air resistance is negligible, and the collision is elastic. The masses of the ball and block are, respectively, 1.60 kg and 2.40 kg, and the length of the wire is 1.20 m. Find the velocity (magnitude and direction) of the ball (a) just before the collision, and (b) just after the collision.

2. Relevant equations
.5m1v2 = mgh
m1v1 + m2v2 = m1vf1 + m2vf2

3. The attempt at a solution
So I've got part a ~
.5m1v2 = mgh
v2 = 2gh

and I'm lost on part b:
m1v1 + m2v2 = m1vf1 + m2vf2
since the final velocities of the ball and box are unknown how do I solve?

2. ### rl.bhat

4,435
After collision, total momentum and the KE is conserved.
Write down equations for these two conservation laws and solve for the final velocities.

3. ### E=mc^84

50
Hi, for a) The ball starts from rest and has potential energy. The potential energy is converted to kinetic energy at the bottom of the string, therefore: mgh = 0.5mv^2, you shud get v = sqrt2gh, where h = the wire legnth = 1.2m.

For b) since the ball collides with the block, initially at rest, you have to use the law of conservation of momentum and adding the block mass and the ball mass and using the speed in the above question as the initial velocity of only the ball: mava = v(mball +mblock), where mava = the speed and mass of the ball respectively from the above question. Therefore, Solve for v.

Hope this helps:)

4. ### jahrollins

25
In mava = v(mball +mblock)
why would the final velocity be the same for both the ball and the block?

5. ### ehild

12,954
The velocities will not be the same after collision.

The collision is elastic, both the momentum and the energy is conserved during the collision.

$$m_1v_{i1}+m_2v_{i2}=m_1v_{f1}+m_2v_{f2}$$

$$\frac{1}{2}m_1v_{i1}^2+\frac{1}{2}m_2v_{i2}^2=\frac{1}{2}m_1v_{f1}^2+\frac{1}{2}m_2v_{f2}^2$$

Just before collision, the block is in rest, vi1=0, and the velocity of the ball is obtained from the equation vi2=sqrt(gL) where the length of wire is L=1.2 m. Calculate vi2 and plug in the values for the initial velocities in both equations.

ehild

6. ### E=mc^84

50
Because they are in contact with each other during the collision for a reallt small amount of time. That velocity cud = the ball or the block since they are both moving at such initial speeds. But, since you are concerned with only the the velocity of the ball, you use only that.

7. ### ehild

12,954
You are wrong. We do not care what happens during that very short time while the colliding bodies are in contact: it is beyond point mechanics. Only the result counts. If the collision is elastic, the kinetic energy is conserved, it is the same before and after collision.

The velocities will be the same if the collision is totally inelastic.

ehild

8. ### E=mc^84

50
You are talking about what you "care about", this is what happens in the problem. Work out the problem and show me that i am wrong :)

9. ### ehild

12,954
Well, do you know what is elastic collision?

ehild

10. ### jahrollins

25
From those two, I came out with vi2 = vf2?

11. ### ehild

12,954
No. Plug in the numbers and solve the system of equations.

ehild


text:
# IEEE Transactions on Automation Science and Engineering

## Issue 1 • Jan. 2018

The purchase and pricing options for this item are unavailable. Select items are only available as part of a subscription package. You may try again later or contact us for more information.

## Filter Results

Displaying Results 1 - 25 of 36

Publication Year: 2018, Page(s):C1 - 1
| |PDF (170 KB)
• ### IEEE Transactions on Automation Science and Engineering

Publication Year: 2018, Page(s): C2
| |PDF (103 KB)
• ### Integration of Learning-Based Testing and Supervisory Control for Requirements Conformance of Black-Box Reactive Systems

Publication Year: 2018, Page(s):2 - 15
| |PDF (2494 KB) | HTML

A fundamental requirement of the supervisory control theory (SCT) of discrete-event systems is a finite automaton model of the plant. The requirement does not hold for black-box systems whose source code and logical model are not accessible. To apply SCT to black-box systems, we integrate automaton learning technology with SCT and apply the new method to improve the requirements conformance of sof... View full abstract»

• ### Asymptotic Tracking and Robustness of MAS Transitions Under a New Communication Topology

Publication Year: 2018, Page(s):16 - 32
| |PDF (3054 KB) | HTML Media

We have recently applied the principles of continuum mechanics to develop a new leader–follower model for the collective motion of a multiagent system (MAS). Agents are modeled as particles of a continuum body that can deform in $mathbb {R}^{n}$ ( $n=1,2,3$ View full abstract»

• ### Optimal Sensor Placement for Monitoring of Spatial Networks

Publication Year: 2018, Page(s):33 - 44
| |PDF (1272 KB) | HTML

The problem of optimal placement of sensors for monitoring a spatial network (e.g., a road network with moving ground targets or intruders) is considered in this paper. In particular, the optimization of locations of a set of sensors (that can each obtain measurements in a local region around the sensor location) is considered so as to maximize an overall sensor coverage metric defined over the sp... View full abstract»

• ### Image-Guided Nanopositioning Scheme for SEM

Publication Year: 2018, Page(s):45 - 56
Cited by:  Papers (1)
| |PDF (8147 KB) | HTML Media

Positioning of micro-nanoobjects inside a scanning electron microscope (SEM) for manipulation is a key and challenging task to perform. Often it is performed by skilled operators via teleoperation, which is tedious and lacks repeatability. In this paper, rendering this task as an image-guided problem, we present a frequency domain scheme for automatic control of positioning platform movements. The... View full abstract»

• ### Continuous Tracking Control for a Compliant Actuator With Two-Stage Stiffness

Publication Year: 2018, Page(s):57 - 66
| |PDF (6164 KB) | HTML

Emerging applications of robots with direct physical interactions with humans have led to the development of a variety of series elastic actuators (SEAs) which are compliant, force controllable, and back drivable. The performance of current SEAs is mainly dependent on the specific stiffness of the spring. In our previous work, a compliant actuator with two-stage stiffness has been designed to over... View full abstract»

• ### A Methodological Approach to Model-Driven Design and Development of Automation Systems

Publication Year: 2018, Page(s):67 - 79
| |PDF (4924 KB) | HTML Media

The growing complexity of industrial automation demands the adoption of software engineering principles for improving the development process of control systems. This paper presents a methodological approach to the design and development of complex automation systems relying on model-driven engineering (MDE). A benefit of this approach is the integration of methods and techniques widespread within... View full abstract»

• ### Simultaneous Multiple-Nanowire Motion Control, Planning, and Manipulation Under Electric Fields in Fluid Suspension

Publication Year: 2018, Page(s):80 - 91
| |PDF (2638 KB) | HTML

To fully take advantage of the enormous potential of functional nanodevices, it is crucial to automate their manufacture with efficient steering and manipulation of multiple nanowires with controlled orientations to specific spatial locations. In this paper, we present motion planning and control algorithms for simultaneously steering multiple nanowires in liquid suspension. The motion planning an... View full abstract»

• ### Event-Based Supervisory Control for Energy Efficient Manufacturing Systems

Publication Year: 2018, Page(s):92 - 103
| |PDF (1507 KB) | HTML

It becomes more and more critical for manufacturing enterprises to improve energy efficiency because of the escalating energy prices, increasing global competitions, and more rigorous government regulations. In this paper, a systematic method is developed to improve the energy efficiency of a multistage manufacturing system through production control. The method aims at reducing energy consumption... View full abstract»

• ### Design of Vector Field for Different Subphases of Gait and Regeneration of Gait Pattern

Publication Year: 2018, Page(s):104 - 110
| |PDF (3086 KB) | HTML

In this paper, we have designed the vector fields (VFs) for all the six joints (hip, knee, and ankle) of a bipedal walking model. The bipedal gait is the manifestation of temporal changes in the six joints angles, two each for hip, knee, and ankle values and it is a combination of seven different discrete subphases. Developing the correct joint trajectories for all the six joints was difficult fro... View full abstract»

• ### Curved Reflection Symmetric Axes on Free-Form Surfaces and Their Extraction

Publication Year: 2018, Page(s):111 - 126
| |PDF (3762 KB) | HTML

Feature detection on smooth free-form surfaces is much more difficult than that on shapes with sharp features. In this paper, we extract the curved reflection axes (CRAs) of an arbitrary free-form surface as features if they exist. The extraction result is robust to boundary noises and strongly sensitive to extrinsic properties of the surface such as projected normals and curvatures. Compared with... View full abstract»

• ### A Spectral Graph Theoretic Approach for Monitoring Multivariate Time Series Data From Complex Dynamical Processes

Publication Year: 2018, Page(s):127 - 144
| |PDF (3700 KB) | HTML

The objective of this paper is to monitor complex process dynamics manifest in multivariate (multidimensional) time series data using a spectral (algebraic) graph theoretic approach. We test the hypothesis that the spectral graph-based topological invariants detect incipient process drifts earlier [lower average run length (ARL1)] and with higher fidelity (consistency of detection) when... View full abstract»

• ### Multifeature, Sparse-Based Approach for Defects Detection and Classification in Semiconductor Units

Publication Year: 2018, Page(s):145 - 159
Cited by:  Papers (3)
| |PDF (3428 KB) | HTML

Automated inspection systems play an important role in manufacturing to guarantee higher quality and reduce production costs. In the semiconductor manufacturing industry, assembly and testing processes are getting more complex, resulting in a greater tendency of defects to impact the production process. These defects can cause field failures and can result in customer dissatisfactions and returns.... View full abstract»

• ### Set-Membership-Based Fault Detection and Isolation for Robotic Assembly of Electrical Connectors

Publication Year: 2018, Page(s):160 - 171
| |PDF (2548 KB) | HTML Media

This paper addresses the fault detection and isolation (FDI) problem for robotic assembly of electrical connectors in the framework of set-membership. Both the fault-free and faulty cases of assembly are modeled by different switched linear models with known switching sequences, bounded parameters, and external disturbances. The locations of switching points of each model are assumed to be inside ... View full abstract»

• ### Analysis and Observations From the First Amazon Picking Challenge

Publication Year: 2018, Page(s):172 - 188
Cited by:  Papers (4)
| |PDF (3918 KB) | HTML

This paper presents an overview of the inaugural Amazon Picking Challenge along with a summary of a survey conducted among the 26 participating teams. The challenge goal was to design an autonomous robot to pick items from a warehouse shelf. This task is currently performed by human workers, and there is hope that robots can someday help increase efficiency and throughput while lowering cost. We r... View full abstract»

• ### Planning and Control for Collision-Free Cooperative Aerial Transportation

Publication Year: 2018, Page(s):189 - 201
Cited by:  Papers (1)
| |PDF (4360 KB) | HTML Media

This paper presents planning and control synthesis for multiple aerial manipulators to transport a common object. Each aerial manipulator that consists of a hexacopter and a two-degree-of-freedom robotic arm is controlled by an augmented adaptive sliding mode controller based on a closed-chain robot dynamics. We propose a motion planning algorithm by exploiting rapidly exploring random tree star (... View full abstract»

• ### Machining-Based Coverage Path Planning for Automated Structural Inspection

Publication Year: 2018, Page(s):202 - 213
Cited by:  Papers (1)
| |PDF (2550 KB) | HTML

The automation of robotically delivered nondestructive evaluation inspection shares many aims with traditional manufacture machining. This paper presents a new hardware and software system for automated thickness mapping of large-scale areas, with multiple obstacles, by employing computer-aided drawing (CAD)/computer-aided manufacturing (CAM)-inspired path planning to implement control of a novel ... View full abstract»

• ### Using Degradation Messages to Predict Hydraulic System Failures in a Commercial Aircraft

Publication Year: 2018, Page(s):214 - 224
| |PDF (1726 KB) | HTML

This paper presents a failure prognostics methodology based on degradation messages using a particle filter framework. In the proposed method, the degradation messages are interpreted as quantized measurements. The use of degradation messages reduces the need for data communication bandwidth, since they only need to be transmitted or stored when some predefined threshold is crossed. In contrast, t... View full abstract»

• ### Data-Defect Inspection With Kernel-Neighbor-Density-Change Outlier Factor

Publication Year: 2018, Page(s):225 - 238
| |PDF (2873 KB) | HTML

Data-defect would affect the data quality and the analysis results of data mining. This paper presents a data-defect inspection method with kernel-neighbor-density-change outlier factor (KNDCOF). The definition of kernel neighbor density is proposed to represent the density of each object in database, and the ascending distance series (ADS) of each object is calculated based on the kernel distance... View full abstract»

• ### Collision Detection and Signal Recovery for UHF RFID Systems

Publication Year: 2018, Page(s):239 - 250
| |PDF (2398 KB) | HTML

In this paper, we present a novel high-throughput anti-collision algorithm for passive ultrahigh-frequency (UHF) radio-frequency identification (RFID) systems. Our algorithm utilizes signal recovery techniques of collided tag signals to both recover tag communications and obtain an accurate count of all tags in the field. Passive UHF RFID systems are used for long-range passive communications for ... View full abstract»

• ### A Noise-Tolerant Algorithm for Robot-Sensor Calibration Using a Planar Disk of Arbitrary 3-D Orientation

Publication Year: 2018, Page(s):251 - 263
| |PDF (2344 KB) | HTML

In a 3-D scanning task, a robot-sensor system controls a robotic arm to move a laser sensor. In order to align the coordinate system of the robotic arm and laser sensor, prior calibration is required to derive the transformation between both coordinate systems. This paper proposes a new calibration method in three steps: manual data collection, sensing data calculation, and transformation solution... View full abstract»

• ### Census Signal Temporal Logic Inference for Multiagent Group Behavior Analysis

Publication Year: 2018, Page(s):264 - 277
Cited by:  Papers (5)
| |PDF (1955 KB) | HTML

In this paper, we define a novel census signal temporal logic (CensusSTL) that focuses on the number of agents in different subsets of a group that complete a certain task specified by the STL. CensusSTL consists of an “inner logic” STL formula and an “outer logic” STL formula. We present a new inference algorithm to infer CensusSTL for... View full abstract»

• ### Efficient Planar Caging Test Using Space Mapping

Publication Year: 2018, Page(s):278 - 289
| |PDF (3112 KB) | HTML Media

This paper presents an efficient algorithm to test whether a planar object can be caged by a formation of point agents (point fingertips or point mobile robots). The algorithm is based on a space mapping between the 2-D work space ( $mathcal {W}$ space) and the 3-D configuration space ( View full abstract»

• ### Steering and Control of Miniaturized Untethered Soft Magnetic Grippers With Haptic Assistance

Publication Year: 2018, Page(s):290 - 306
Cited by:  Papers (1)
| |PDF (5896 KB) | HTML Media

Untethered miniature robotics have recently shown promising results in several scenarios at the microscale, such as targeted drug delivery, microassembly, and biopsy procedures. However, the vast majority of these small-scale robots have very limited manipulation capabilities, and none of the steering systems currently available enables humans to intuitively and effectively control dexterous minia... View full abstract»

## Aims & Scope

T-ASE will publish foundational research on Automation: scientific methods and technologies that improve efficiency, productivity, quality, and reliability, specifically for methods, machines, and systems operating in structured environments over long periods, and the explicit structuring of environments.

Full Aims & Scope

## Meet Our Editors

Editor-in-Chief
Michael Y. Wang
HKUST Robotics Institute
Hong Kong University of Science and Technology


text:
# Managing conda virtual environments and installing extra packages¶

You may want to install additional python packages which are not bundled with the standard IPKISS installation. This may for instance be the case if you do a lot of data processing (e.g. pandas) or want to run automated regression tests (e.g. pytest).

Most mainstream packages such as pandas and pytest are compatible with IPKISS, but sometimes you may need to install packages of which you’re unsure if they risk breaking IPKISS. For this reason, we advise to create a separate virtual environment (virtualenv or venv in short). IPKISS is based on the conda (www.anaconda.org) virtual environment and package manager, which allows to easily create your own virtual environments (conda environments) with their own set of packages. This enables shielding of your test and development environments from your main IPKISS environment.

In fact, IPKISS runs in its own virtual environment. The main ipkiss environment is named ipkiss3. You can see this when running the IPKISS Terminal from the Luceda Control Center:

IPKISS Terminal just after starting, showing the current virtual environment ipkiss3.

You can list the available environments by running the following conda command in the IPKISS terminal:

conda info --envs

A more complete reference can be found on the conda documentation webpage.

If this is the first time you installed IPKISS or Anaconda on your computer, this only prints the ipkiss3 environment (where 330 is replaced by the ipkiss version number and C:luceda by the actual installation location on your disk).

ipkiss3         *   C:\luceda\ipkiss_330\python\envs\ipkiss3

## Creating a development environment¶

### Step 1: clone¶

We advise to create a new ipkiss conda environment by cloning (copying) the default ipkiss3 environment. This can be done with the conda create command from the IPKISS terminal, supplying a name for the new virtual environment (e.g. datascience in the example below) and the name of the environment to clone (ipkiss3):

conda create -n datascience --clone ipkiss3.

Creating a new virtualenv named datascience by cloning the ipkiss3 environment.

When successful, conda info –envs now lists your newly created environment. If you have a valid license file installed in your IPKISS installation, it will automatically be copied to the new virtualenv as well.

### Step 2: activate environment¶

After cloning the ipkiss3 environment, you need to activate your newly created environment in order to work with it:

conda activate datascience

This makes sure that the correct python will be run and the correct set of packages will be used. Each time you want to work with this environment, just run activate <environment name> as above.

When using a Python Integrated Development Environment (IDE) such as PyCharm, you will need to configure the correct interpreter (python.exe inside your new conda environment) as well. This is explained in detail here in the Python IDE tutorial.

### Step 3: test¶

Quickly test your new IPKISS environment by running import ipkiss3.all as i3 with Python:

python -c "import ipkiss3.all as i3"

This should print that you have a valid license:

## Installing extra packages¶

Now you can install extra packages in the new virtual environment. As mentioned earlier, this will not disturb the original ipkiss3 environment. As an example, we can install pandas (https://pandas.pydata.org), a package popular in data science, with the following command:

conda install pandas

While running the installation, you may get a few warnings which you can ignore, and you will need to confirm the installation by pressing ‘y’:


text:
# Exercise 11.2.2

Demonstrate what happens when we insert the keys 5, 28, 19, 15, 20, 33, 12, 17, 10 into a hash table with collisions resolved by chaining. Let the table have 9 slots, and let the hash function be $h(k) = k \bmod 9$.

First, let's calculate the hashes:

h(5)  = 5
h(28) = 1
h(19) = 1
h(15) = 6
h(20) = 2
h(33) = 6
h(12) = 3
h(17) = 8
h(10) = 1

Next, let's ASCII-ART this bad boy:

  +-----+
0 |     |
+-----+
1 |  o--|---> [ 10 ] ---> [ 19 ] ---> [ 28 ]
+-----+
2 |  o--|---> [ 20 ]
+-----+
3 |  o--|---> [ 12 ]
+-----+
4 |     |
+-----+
5 |  o--|---> [  5 ]
+-----+
6 |  o--|---> [ 33 ] ---> [ 15 ]
+-----+
7 |     |
+-----+
8 |  o--|---> [ 17 ]
+-----+

Where each cell of the array is a null pointer (empty bucket) or the pointer to a head of a linked list.


text:
# Fractional Limit

Calculus Level 4

$\large \displaystyle \lim_{x \rightarrow 8} \dfrac{\sin\{x - 10\}}{\{10 - x\}}$

Which are the following are true for the limit above?

Details :
$$\{ . \}$$ denotes fractional part of x .
LHL and RHL means Left hand limit and Right hand limit respectively.

×


text:
# American Institute of Mathematical Sciences

January  2002, 8(1): 1-15. doi: 10.3934/dcds.2002.8.1

## On random Schrödinger operators on $\mathbb Z^2$

 1 Institute for Advanced Study, 1 Einstein Drive, Princeton, New Jersey 08540

Received  October 2001 Published  October 2001

This paper is concerned with random lattice operators on $\mathbb Z^2$ of the form $H_\omega = \Delta + V_\omega$ where $\Delta$ is the lattice Laplacian and $V_\omega$ a random potential $V_\omega(n) = \omega_nv_n, \{\omega_n\}$ independent Bernoulli or Gaussian variables and $\{v_n\}$ satisfying the condition sup$_n |v_n| |n|^\rho < \infty$ for some $\rho>\frac{1}{2}$. In this setting and restricting the spectrum away from the edges and 0, existence and completeness of the wave operators is shown. This leads to statements on the a.c. spectrum of $H_\omega$. It should be pointed out that, although we do consider here only a specific (and classical) model, the core of our analysis does apply in greater generality.
Citation: Jean Bourgain. On random Schrödinger operators on $\mathbb Z^2$. Discrete & Continuous Dynamical Systems - A, 2002, 8 (1) : 1-15. doi: 10.3934/dcds.2002.8.1
 [1] C. T. Cremins, G. Infante. A semilinear $A$-spectrum. Discrete & Continuous Dynamical Systems - S, 2008, 1 (2) : 235-242. doi: 10.3934/dcdss.2008.1.235 [2] Qiongwei Huang, Jiashi Tang. Bifurcation of a limit cycle in the ac-driven complex Ginzburg-Landau equation. Discrete & Continuous Dynamical Systems - B, 2010, 14 (1) : 129-141. doi: 10.3934/dcdsb.2010.14.129 [3] Dmitry Dolgopyat, Dmitry Jakobson. On small gaps in the length spectrum. Journal of Modern Dynamics, 2016, 10: 339-352. doi: 10.3934/jmd.2016.10.339 [4] Natalija Sergejeva. On the unusual Fucik spectrum. Conference Publications, 2007, 2007 (Special) : 920-926. doi: 10.3934/proc.2007.2007.920 [5] Umesh V. Dubey, Vivek M. Mallick. Spectrum of some triangulated categories. Electronic Research Announcements, 2011, 18: 50-53. doi: 10.3934/era.2011.18.50 [6] Tim Alderson, Alessandro Neri. Maximum weight spectrum codes. Advances in Mathematics of Communications, 2019, 13 (1) : 101-119. doi: 10.3934/amc.2019006 [7] J. Douglas Wright. On the spectrum of the superposition of separated potentials.. Discrete & Continuous Dynamical Systems - B, 2013, 18 (1) : 273-281. doi: 10.3934/dcdsb.2013.18.273 [8] David Damanik, Anton Gorodetski. The spectrum of the weakly coupled Fibonacci Hamiltonian. Electronic Research Announcements, 2009, 16: 23-29. doi: 10.3934/era.2009.16.23 [9] Yves Guivarc'h. On the spectrum of a large subgroup of a semisimple group. Journal of Modern Dynamics, 2008, 2 (1) : 15-42. doi: 10.3934/jmd.2008.2.15 [10] Frédéric Naud. The Ruelle spectrum of generic transfer operators. Discrete & Continuous Dynamical Systems - A, 2012, 32 (7) : 2521-2531. doi: 10.3934/dcds.2012.32.2521 [11] Nazar Arakelian, Saeed Tafazolian, Fernando Torres. On the spectrum for the genera of maximal curves over small fields. Advances in Mathematics of Communications, 2018, 12 (1) : 143-149. doi: 10.3934/amc.2018009 [12] Gary Froyland. On Ulam approximation of the isolated spectrum and eigenfunctions of hyperbolic maps. Discrete & Continuous Dynamical Systems - A, 2007, 17 (3) : 671-689. doi: 10.3934/dcds.2007.17.671 [13] Philipp Kunde. Smooth diffeomorphisms with homogeneous spectrum and disjointness of convolutions. Journal of Modern Dynamics, 2016, 10: 439-481. doi: 10.3934/jmd.2016.10.439 [14] Frédéric Naud. Birkhoff cones, symbolic dynamics and spectrum of transfer operators. Discrete & Continuous Dynamical Systems - A, 2004, 11 (2&3) : 581-598. doi: 10.3934/dcds.2004.11.581 [15] Michael Khanevsky. Hofer's length spectrum of symplectic surfaces. Journal of Modern Dynamics, 2015, 9: 219-235. doi: 10.3934/jmd.2015.9.219 [16] Alexei A. Ilyin. Lower bounds for the spectrum of the Laplace and Stokes operators. Discrete & Continuous Dynamical Systems - A, 2010, 28 (1) : 131-146. doi: 10.3934/dcds.2010.28.131 [17] Dmitry Jakobson, Alexander Strohmaier, Steve Zelditch. On the spectrum of geometric operators on Kähler manifolds. Journal of Modern Dynamics, 2008, 2 (4) : 701-718. doi: 10.3934/jmd.2008.2.701 [18] Vladimir Müller, Aljoša Peperko. On the Bonsall cone spectral radius and the approximate point spectrum. Discrete & Continuous Dynamical Systems - A, 2017, 37 (10) : 5337-5354. doi: 10.3934/dcds.2017232 [19] Lucia D. Simonelli. Absolutely continuous spectrum for parabolic flows/maps. Discrete & Continuous Dynamical Systems - A, 2018, 38 (1) : 263-292. doi: 10.3934/dcds.2018013 [20] M. C. Carbinatto, K. Mischaikow. Horseshoes and the Conley index spectrum - II: the theorem is sharp. Discrete & Continuous Dynamical Systems - A, 1999, 5 (3) : 599-616. doi: 10.3934/dcds.1999.5.599

2018 Impact Factor: 1.143


text:
# Application Gallery

 Navigation: » No topics above this level « Inverse design of waveguide crossing

Using parametric optimization (PO), and python interoperability design a silicon-on-insulator waveguide crossing in varFDTD 2.5D and 3D FDTD to rapidly converge on a solution.  This optimization algorithm is based on the adjoint method, and yields optimal solution in just a few iterations compared to other schemes such as particle swarm optimization (PSO). The final design will then be exported into a GDS file for further simulation and/or fabrication.

### Files and Required Products

Minimum product version: 2019b r3

### Contents

Overview

Run and results

Important model settings

Updating the model with your parameters

## Overview

Understand the simulation workflow and key results

Lumerical's inverse design capability provides unparalleled optimization performance by combining the power of gradient based optimization routine with efficiencies found in fundamental properties of Maxwell equations.

This example will demonstrate how to use the inverse design method to generate a crossing shape that delivers near perfect transmission efficiency. Moreover, we'll demonstrate how to modify the example with your parameters so you can reuse this approach for your own design.

This example draws extensively from the LumOpt framework:

Git

### Step 1: Define base simulation parameters

The goal for this initial step is to define the basic parameters of the FDTD simulation that will be used for the adjoint optimization. Here, key simulation settings for the crossing waveguides, monitors for field gradient and figure of merit (FOM) calculations, source and mesh are parametrized for ease of running the optimization routine. It is possible to provide these necessary parameters with either

oMODE project file

olumerical script (.lsf format)

oPython function handle.

In this example we will use a lumerical script file to set up the simulation that works with the LumOpt optimization routines.

### Step 2: Define the optimizable geometry

The second step is to define a parametrized polygon geometry that will represent the body of the crossing connecting the input and output waveguides (the optimizable geometry). This is defined as a python function that accepts a set of optimization parameters and generates corresponding polygon geometry appended to the base FDTD simulation geometry defined in step 1. Initial parameter values and their limits will also get defined in this step.

### Step 3: Run the parametric optimization in 2.5D

Run the python script with the base simulation script and parametrized polygon object as an input. This will be a 2.5D simulation for rapid delivery of an optimal design which can be used as the initial guess for the 3D optimization in the next step. To find the best set of parameters that define the optimal crossing shape, the optimization routine will use the specified field monitors in the base simulation to calculate:

The field gradients: the gradients of the field due to perturbation of permittivity resulting from slight changes in geometry of the optimizable region

The gradient of the figure of merit (FOM): the gradient of the mode overlap to the fundamental TE mode of the waveguides as a result of the shape changes

After optimization, the optimized crossing component shape will be exported into GDS II format which can be used for further simulation and/or fabrication (mask design).

### Step 4: Run the parametric optimization in 3D [optional]

As an optional step, the best solution found by the 2.5D optimization can be used as an initial guess for a 3D optimization to verify the 2.5D simulation results and to further improve the performance of the design if possible. If one does not have a varFDTD license it is possible to start with the 3D optimization; however, without an initial solution the optimization may progress slowly. We have included symmetric boundary conditions to reduce the sim time by a factor of four.

## Run and results

Instructions for running the model and discussion of key results

### Step 1: Define base simulation parameters

This step is not necessary if you want to run the example as is, however it is required for modification of base simulation parameters based on your needs.

1.Open the base simulation script file (varFDTD_crossing.lsf) in FDTD solutions

2.Modify the desired base simulation parameters

3.Save and run the script file to make sure the base simulation is setup as intended and the script runs successfully. This is just a sanity check and the generated simulation is not required to be saved for the next step

### Base simulation setup

The resulting base simulation which includes the FDTD simulation region, source, monitors, mesh override for optimizable geometry, and input and output waveguides is shown in the figure below.

### Step 2: Define the optimizable geometry

This step is not necessary if you want to run the example as is, however it is required for modification of base simulation parameters based on your needs.

1.Open the parametric optimization python script file (crossing_opt_varFDTD.py) in FDTD script editor

2.Define the initial values of the input parameters of the optimization (the Y coordinates of the nodes of the optimizable polygon) through the array initial_params

3.Define the bounds of the optimization parameters within the array bounds

4.Make any other necessary modifications to the optimization parameters. See “Important model settings” for more information about these parameters

5.Save the script

### Optimization python script

Below is a screenshot of the optimization python script within the script editor window in FDTD

### Step 3: Run the parametric optimization in 2D

1.Before running the optimization, make sure the current working directory for FDTD is set to the location where the example files are located

2.Run the optimization script file from the script editor in FDTD. Alternatively, you can run the optimization script from the command line using the Python executable provided with the Lumerical installation.

### 2D optimization results

The geometry of the crossing at the start of the optimization with default parameters is shown in the figure below:

A command prompt (in windows operating system) or terminal (in macOS or Linux) window will open showing the progress of the optimization. A screenshot of the command prompt window at the end of the optimization is shown below which contains information about the optimization progress

During the optimization, a new session of varFDTD will be opened and closed by the optimization routine to perform simulations on the guesses made by the optimization algorithm. Another window will also report the optimization progress providing various plots for evolution of the optimization geometry, figure of merit, parameters and other related information. These CAD windows can be hidden by specifying 'hide_fdtd_cad=true' in the optimizer. Another window will also report the optimization progress providing various plots for evolution of the optimization geometry, figure of merit, parameters and other related information. The figure below shows the optimization report window at the end of optimization process.  The geometry of the final solution can be seen in the plot and it is shown that the FOM will have little improvement after about 25 iterations which suggests the number of iterations can be further reduced for future runs. The field gradients (due to the changes in geometry) and forward direction field distributions within the device’s structure can also be seen in the plot. In addition, it is clear that the changes in the optimized parameter values become less intense after a few iterations since the FOM gets closer to the optimum value. The changes in FOM gradient (based on changes in the parameter values) also decrease on average as seen in gradient evolution plot as the iteration continues.

From the plot, it can be seen that the optimization started with an initial design providing an FOM of ~0.75 and ended up with a geometry with a FOM of about 0.96, resulting in about 27 % improvement through the optimization process in just a few iterations which highlights the advantage of this method compared to other optimization algorithms such as particle swarm which need many iterations to achieve an optimum solution. Also, note that FOM will have little improvement after about 5 iterations which suggests higher number of iterations probably won't achieve a significant improvement. The field gradients (due to the changes in geometry) and forward direction field distributions within the device’s structure can also be seen in the plot. In addition, the changes in the optimizable parameter values become less noticeable after a few iterations as the FOM gets closer to the optimum value. The changes in FOM gradient (based on changes in the parameter values) also decrease on average as seen in gradient evolution plot as a function of the iterations.

At the end of optimization, the final parameter values will be saved in a text file (2D_parameters.txt) in the same folder as the example files which will be used in the 3D optimization for the next step. In addition, copies of the simulation files for each iteration, a text file containing the history of parameters and FOM values for each iteration and still images of the optimization report window for each iteration will be saved in a folder (with a name starting with “opts”) in the same folder as example files for future reference. Each optimization run creates a new folder with the same name but a numbered suffix to avoid overwriting previous runs data.

Here is the final geometry optimized in 2D:

The optimization python script file will also export the optimal design to a GDS file at the end of optimization. The GDS file will be saved in the same folder as the example files. The example files come with an encrypted script files with .lsfx extension that is needed for the GDS export operation. They should be saved in the same location as the python script file.

### Step 4: Run the parametric optimization in 3D (optional)

1.Before running the optimization, make sure the current working directory for FDTD is set to the location where the example files are located

2.Make sure the base simulation script file (FDTD_crossing.lsf) is setup correctly

3.Open and run the optimization python script file (crossing_opt_3D.py) in FDTD script editor. Alternatively, you can run the optimization script from the command line using the Python executable provided with the Lumerical installation. The script will read the final parameters from the 2D optimization (stored in 2D_parameters.txt) if available. Otherwise, it will start with default values set in the script.

### 3D optimization results

Note that the 3D optimization might take a long time to complete due to the large size of the simulation.

Looking at the results of the 3D optimization will reveal that the crossing’s transmission has been initially predicted to be about 82% by 3D simulation (which is more realistic than the 2D case) and has been improved to 91% at the end of optimization. It can be seen that the FOM does not improve significantly after 6 iterations suggesting that running a longer optimization may not result in much better performance.

## Important model settings

Description of important objects and settings used in this model

Definition of optimizable geometry: The optimizable geometry is defined as a polygon which has fixed values for the x coordinates of its boundary points and the points’ y coordinates can be modified to obtain the optimal geometry. These y coordinates are defined as the nodes of a cubic spline (as shown below), which makes it possible to perform cubic spline interpolation on the points in order to obtain a smooth polygon geometry. For the waveguide crossing, this optimizable polygon needs to be mirrored rotationally in order to form a complete crossing, meaning that only 1/8 of the optimizable geometry is independently parametrized. It is important to make sure that the dimensions of the optimizable geometry and the waveguides set in the base simulation match so that the crossing geometry is properly formed.

Optimization fields monitor: The optimization fields DFT monitor (opt_field) is used to collect the field data within the optimizable geometry which is then used to calculate field gradients used in optimization algorithm. As such, the location of this monitor is of great importance and should cover the entire optimizable geometry.

Figure of merit fields monitor: Since this monitor is used to calculate the figure of merit required for optimization (mode overlap to the fundamental TE mode of the output waveguide), it should be located within the output waveguide of the crossing with appropriate dimensions.

Height of the geometry objects: The height (depth) of the geometry objects should match in both base simulation (waveguides) and the optimizable geometry (polygon). For the waveguides, this is set through the base simulation setup script (by setting their z span) and for the polygon, it is adjusted as the input argument “depth” when calling the cross function to form the optimizable geometry. This value should also be passed to the GDS export portion of the script as depth:

Optimization parameters bounds and initial values: The range over which the optimization algorithm is allowed to vary the parameters and also the initial values for those parameters as a starting point are defined by initial_params and bounds arrays as input arguments when calling the function FunctionDefinedPolygon

Simulation bandwidth: The inverse design PO method supports broadband simulations. The bandwidth is defined using start and stop wavelengths complemented by the number of frequency points in the given interval. The wavelength specification is passed to the optimization script when defining the FOM. Therefore, the wavelength settings defined in the base simulation script will be overridden.

Additionally, each frequency point can have their target FOM specified independently. This means that the optimization routine can aim to match custom FOM profile as a function of wavelength rather than simply maximizing the FOM across the full bandwidth.

Here is a code section from this Y-branch example that specifies the bandwidth from 1300 to 1800nm with 21 frequency points:

wavelengths = Wavelengths(start = 1300e-9, stop = 1800e-9, points = 21)

The bandwidth settings is then passed to the optimization routine during FOM definition. Notice that we specified FOM target of 1 for all frequency points to maximize the transmission across the desired bandwidth:

fom = ModeMatch(monitor_name = 'fom',

mode_number = 'fundamental mode',

direction = 'Forward',

target_T_fwd = lambda wl: np.ones(wl.size),

norm_p = 1)

Maximum number of iterations: While the algorithm has the capability to stop the optimization once the gradient of the figure of merit falls below a certain threshold, the “max_iter” variable used for defining the optimization algorithm can be used to limit the number of iterations the algorithm is able to perform.

Symmetry: As of 2019a r4 the inverse design PO method supports symmetric boundaries. This can be particularly useful for 3D simulations where it can decrease the simulation time by factor of 4.

## Updating the model with your parameters

Instructions for updating the model based on your device parameters

Geometry: If you need to have your own geometry defined for the crossing, including SOI device layer thickness (waveguide height) and waveguide width, corresponding changes should be made in base simulation setup script and/or optimization setup python script. This might need changes in objects span and location in the base script, the polygon’s “depth” parameter in python script which its value is also passed to the GDS export portion of the script. The source, simulation region, mesh override, and field gradient and fom monitors dimensions should also be adjusted accordingly to make sure they cover the entire structure properly. The coordinates of the initial values and bounds of the optimizable geometry at points where it is connected to the waveguide might need to be adjusted to ensure proper connection between the optimizable region and the waveguides.

def cross(params):

y_end = params[-1]

x_end = 0 - y_end

points_x = np.concatenate(([-2.01e-6], np.linspace(-2e-6, x_end, 10)))

points_y = np.concatenate(([0.25e-6], params))

n_interpolation_points = 50

polygon_points_x = np.linspace(min(points_x), max(points_x), n_interpolation_points)

interpolator = sp.interpolate.interp1d(points_x, points_y, kind = 'cubic')

polygon_points_y = [max(min(point, 1e-6), -1e-6) for point in interpolator(polygon_points_x)]

pplu = [(x, y) for x, y in zip(polygon_points_x, polygon_points_y)]

ppld = [(x, -y) for x, y in zip(polygon_points_x, polygon_points_y)]

ppdl = [(-y, x) for x, y in zip(polygon_points_x, polygon_points_y)]

ppdr = [(y, x) for x, y in zip(polygon_points_x, polygon_points_y)]

pprd = [(-x, -y) for x, y in zip(polygon_points_x, polygon_points_y)]

ppru = [(-x, y) for x, y in zip(polygon_points_x, polygon_points_y)]

ppur = [(y, -x) for x, y in zip(polygon_points_x, polygon_points_y)]

ppul = [(-y, -x) for x, y in zip(polygon_points_x, polygon_points_y)]

polygon_points = np.array(pplu[::-1] + ppld[:-1] + ppdl[::-1] + ppdr[:-1] + pprd[::-1] + ppru[:-1] + ppur[::-1] + ppul[:-1])

return polygon_points

It is recommended to set the coordinates such that there is a slight overlap between the optimizable region and the edges of the waveguides to ensure they are connected in the simulation.

Materials: The materials included in the simulation (the material making the optimizable geometry and the material surrounding it) are defined in Step 1 via lsf, project file or python function. In this example we have taken the index of silicon and silicon dioxide at the desired simulation wavelength of 1550nm. One should make sure that the material defined (or given index) in the initialization is the same as that passed to the optimizer when defining the geometry as shown below;

eps_in = Material(name = 'Si: non-dispersive', mesh_order = 2)

eps_out = Material(name = 'SiO2: non-dispersive', mesh_order = 3)

depth = 220.0e-9

polygon = FunctionDefinedPolygon(func = splitter,

initial_params = initial_points_y,

bounds = bounds,

z = 0.0,

depth = depth,

eps_out = eps_out,

eps_in = eps_in,

edge_precision = 5,

dx = 1.0e-13)

With varFDTD it is now possible to use the material library to define the material properties; however, the optimization technique has not been tested with dispersive materials; thus, the effective index is solved at the center frequency and passed to the optimizer.

Additional documentation, examples and training material

KX posts with tips and related content

## Appendix: Parametric Optimization (Adjoint method)

In a typical device optimization, the goal is to minimize or maximize a figure or merit $$F(p)$$ that depends on a set of $$N$$ design parameters $$p=(p_1,...,p_n)$$. The figure of merit can be any quantity used to benchmark the device’s performance and the design parameters can be any quantity that alters the device’s response. To evaluate $$F(p)$$ for a given set of values of $$p$$, a physics based modeling tool - such as FDTD - is used to map values of $$p$$ to values of $$F$$. The model usually contains a description of the device’s geometry, materials and all relevant environmental conditions. In a parameterized shape optimization, the goal is to allow some parts of the device’s geometry to vary to minimize or maximize the figure of merit. For this purpose, a shape function $$f(p)$$ is typically defined to specify part of the device’s geometry; an example is shown in the following figure:

Y-splitter with a parameterized junction boundary $f(p)$ defined using connected splines.

The shape function is typically user defined and must be fed to the modelling tool to construct the device’s geometry. Once the figure of merit $$F(f(p))$$ can be evaluated for a given shape function, a minimizer can make multiple evaluations of the figure of merit to find the optimal shape. There is a very large body of minimization tools and techniques that can be used to try to find the optimal shape parameters. When selecting a minimization technique, it is important to consider that running a device simulation to evaluate the figure of merit can be a slow operation. So, the best suited minimization techniques are those that limit the number of evaluations of the figure of merit. With a good starting point, gradient based methods can quickly find the optimal parameters with a relatively small number of figure of merit evaluations provided that it is also possible to evaluate the gradient of the figure of merit:

$$∇_pF=(\frac{\partial F}{\partial p_1},⋯,\frac{\partial F}{\partial p_n})$$

To evaluate the gradient without having to perform one simulation for each partial derivative, a technique called the adjoint method can be employed. This technique exploits the properties of linear partial differential equations to evaluate the gradient using only two simulations independently of the number of optimization parameters. The Python based shape optimization module in FDTD (lumopt) allows users to define a shape function and run the corresponding device simulation to evaluate the figure of merit. The module also implements the adjoint method to provide the gradient. In this way, any of the gradient based minimizers in SciPy’s optimize module can be used to perform a fully automated optimization.

Copyright Lumerical Inc. | Privacy | Site Map