text: Students are asked to simplifying 18 radical expressions some containing variables and negative numbers there are 3 imaginary numbers. If n is odd, and b â  0, then. The quotient rule states that a radical involving a quotient is equal to the quotients of two radicals. This problem does not contain any errors; . The two radicals that are being multiplied have the same root (3), so they can be multiplied together underneath the same radical sign. A Variable is a symbol for a number we don't know yet. You correctly took the square roots of. A) Correct. We just have to work with variables as well as numbers. Correct. Simplify each expression by factoring to find perfect squares and then taking â¦ When dividing radical expressions, the rules governing quotients are similar: . This problem does not contain any errors. The simplified form is . Recall that the Product Raised to a Power Rule states that . The conjugate of is . Like Radicals : The radicals which are having same number inside the root and same index is called like radicals. (Express your answer in simplest radical form) How would the expression change if you simplified each radical first, before multiplying? Since all the radicals are fourth roots, you can use the rule Â to multiply the radicands. Multiplying And Dividing Radicals Worksheets admin April 22, 2020 Some of the worksheets below are Multiplying And Dividing Radicals Worksheets, properties of radicals, rules for simplifying radicals, radical operations practice exercises, rationalize the denominator and multiply with radicals worksheet with â¦ Quiz: Dividing Rational Expressions Adding and Subtracting Rational Expressions Examples of Rational Expressions So, this problem and answer pair is incorrect. This algebra video tutorial explains how to multiply radical expressions with variables and exponents. 1) Factor the radicand (the numbers/variables inside the square root). This is an advanced look at radicals. Drop me an email if you have any specific questions. According to the Product Raised to a Power Rule, this can also be written , which is the same as , since fractional exponents can be rewritten as roots. This problem does not contain any errors; You can use the same ideas to help you figure out how to simplify and divide radical expressions. Quiz & Worksheet - Dividing Radical Expressions | Study.com #117518 Answer D contains a problem and answer pair that is incorrect. Multiplying and dividing radicals. As long as the roots of the radical expressions are the same, you can use the Product Raised to a Power Rule to multiply and simplify. What can be multiplied with so the result will not involve a radical? If these are the same, then â¦ Free printable worksheets with answer keys on Radicals, Square Roots (ie no variables)includes visual aides, model problems, exploratory activities, practice problems, and an online component What if you found the quotient of this expression by dividing within the radical first, and then took the cube root of the quotient? The same is true of roots. So, for the same reason that , you find that . In both cases, you arrive at the same product, . simplifying radicals with variables examples, LO: I can simplify radical expressions including adding, subtracting, multiplying, dividing and rationalizing denominators. Are you sure you want to remove #bookConfirmation# Remember that when an exponential expression is raised to another exponent, you multiply â¦ We factor, find things that are squares (or, which is the same thing, find factors that occur in pairs), and then we pull out one copy of whatever was squared (or of whatever we'd found a pair of). For the purpose of the examples below, we are assuming that variables in radicals are non-negative, and denominators are nonzero. You can multiply and divide them, too. Use the rule Â to create two radicals; one in the numerator and one in the denominator. This worksheet correlates with the 1 2 day 2 simplifying radicals with variables power point it contains 12 questions where students are asked to simplify radicals that contain variables. The number coefficients are reduced the same as in simple fractions. Factor the number into its prime factors and expand the variable(s). Look for perfect cubes in the radicand, and rewrite the radicand as a product of factors. Variables with Exponents How to Multiply and Divide them What is a Variable with an Exponent? Since, Identify and pull out powers of 4, using the fact that, Since all the radicals are fourth roots, you can use the rule, Now that the radicands have been multiplied, look again for powers of 4, and pull them out. ... Equations for calculating, algebra 2 practice tests, radicals with variables. You correctly took the square roots of Â and , but you can simplify this expression further. This problem does not contain any errors; . You have applied this rule when expanding expressions such as (. © 2020 Houghton Mifflin Harcourt. Since both radicals are cube roots, you can use the rule Â to create a single rational expression underneath the radical. Incorrect. That choice is made so that after they are multiplied, everything under the radical sign will be perfect cubes. A worked example of simplifying an expression that is a sum of several radicals. Look at the two examples that follow. Answer D contains a problem and answer pair that is incorrect. When dividing variables, you write the problem as a fraction. We can add and subtract like radicals â¦ You simplified , not . Since both radicals are cube roots, you can use the rule, As you become more familiar with dividing and simplifying radical expressions, make sure you continue to pay attention to the roots of the radicals that you are dividing. Free math notes on multiplying and dividing radical expressions. Be looking for powers of 4 in each radicand. A) Problem: Â Answer: 20 Incorrect. But you canât multiply a square root and a cube root using this rule. Answer D contains a problem and answer pair that is incorrect. The "n" simply means that the index could be any value.Our examples will be using the index to be 2 (square root). Module 4: Dividing Radical Expressions Recall the property of exponents that states that m m m a a b b ââ =ââ ââ . So I'll simplify the radicals first, and then see if I can go any further. You can do more than just simplify radical expressions. The terms in this expression are both cube roots, but I can combine them only if they're the cube roots of the same value. When you add and subtract variables, you look for like terms, which is the same thing you will do when you add and subtract radicals. Definition: If $$a\sqrt b + c\sqrt d$$ is a radical expression, then the conjugate is $$a\sqrt b - c\sqrt d$$. I usually let my students play in pairs or groups to review for a test. Correct. Quiz Dividing Radical Expressions. Which one of the following problem and answer pairs is incorrect? An exponent (such as the 2 in x 2) says how many times to use the variable in a multiplication. Multiplying and dividing radical expressions worksheet with answers Collection. Adding and subtracting radicals is much like combining like terms with variables. from your Reading List will also remove any What if you found the quotient of this expression by dividing within the radical first, and then took the cube root of the quotient? As you can see, simplifying radicals that contain variables works exactly the same way as simplifying radicals that contain only numbers. This worksheet has model problems worked out, step by step as well as 25 scaffolded questions that start out relatively easy and end with some real challenges. The same is true of roots: . Then, using the greatest common factor, â¦ That's a mathematical symbols way of saying that when the index is even there can be no negative number in the radicand, but when the index is odd, there can be. Using the Product Raised to a Power Rule, you can take a seemingly complicated expression. It is usually a letter like x or y. In both problems, the Product Raised to a Power Rule is used right away and then the expression is simplified. Answer D contains a problem and answer pair that is incorrect. Use the Quotient Raised to a Power Rule to rewrite this expression. Whichever order you choose, though, you should arrive at the same final expression. For example, while you can think of as equivalent to since both the numerator and the denominator are square roots, notice that you cannot express as . ©o 6KCuAtCav QSMoMfAtIw0akrLeD nLrLDCj.r m 0A0lsls 1r6i4gwh9tWsx 2rieAsKeLrFvpe9dc.c G 3Mfa0dZe7 UwBixtxhr AIunyfVi2nLimtqel bAmlCgQeNbarwaj w1Q.V-6-Worksheet by Kuta Software LLC Answers to Multiplying and Dividing Radicals Slopes of Parallel and Perpendicular Lines, Quiz: Slopes of Parallel and Perpendicular Lines, Linear Equations: Solutions Using Substitution with Two Variables, Quiz: Linear Equations: Solutions Using Substitution with Two Variables, Linear Equations: Solutions Using Elimination with Two Variables, Quiz: Linear Equations: Solutions Using Elimination with Two Variables, Linear Equations: Solutions Using Matrices with Two Variables, Linear Equations: Solutions Using Graphing with Two Variables, Quiz: Linear Equations: Solutions Using Graphing with Two Variables, Quiz: Linear Equations: Solutions Using Matrices with Two Variables, Linear Equations: Solutions Using Determinants with Two Variables, Quiz: Linear Equations: Solutions Using Determinants with Two Variables, Linear Inequalities: Solutions Using Graphing with Two Variables, Quiz: Linear Inequalities: Solutions Using Graphing with Two Variables, Linear Equations: Solutions Using Matrices with Three Variables, Quiz: Linear Equations: Solutions Using Matrices with Three Variables, Linear Equations: Solutions Using Determinants with Three Variables, Quiz: Linear Equations: Solutions Using Determinants with Three Variables, Linear Equations: Solutions Using Elimination with Three Variables, Quiz: Linear Equations: Solutions Using Elimination with Three Variables, Quiz: Trinomials of the Form x^2 + bx + c, Quiz: Trinomials of the Form ax^2 + bx + c, Adding and Subtracting Rational Expressions, Quiz: Adding and Subtracting Rational Expressions, Proportion, Direct Variation, Inverse Variation, Joint Variation, Quiz: Proportion, Direct Variation, Inverse Variation, Joint Variation, Adding and Subtracting Radical Expressions, Quiz: Adding and Subtracting Radical Expressions, Solving Quadratics by the Square Root Property, Quiz: Solving Quadratics by the Square Root Property, Solving Quadratics by Completing the Square, Quiz: Solving Quadratics by Completing the Square, Solving Quadratics by the Quadratic Formula, Quiz: Solving Quadratics by the Quadratic Formula, Quiz: Solving Equations in Quadratic Form, Quiz: Systems of Equations Solved Algebraically, Quiz: Systems of Equations Solved Graphically, Systems of Inequalities Solved Graphically, Systems of Equations Solved Algebraically, Quiz: Exponential and Logarithmic Equations, Quiz: Definition and Examples of Sequences, Binomial Coefficients and the Binomial Theorem, Quiz: Binomial Coefficients and the Binomial Theorem, Online Quizzes for CliffsNotes Algebra II Quick Review, 2nd Edition. Look for perfect cubes can rewrite the expression by a fraction such as ( simplified each radical,! Concerning multiplying and dividing radical dividing radicals with variables that, you arrive at the same manner adding, subtracting multiplying! Is known as the conjugate by another square root multiplying the expression by a fraction having value. While you can rewrite the radicand as the product Raised to a Power rule and! If one student in the numerator is a square root, you can simplify this square root divided by square! With this title is accomplished by multiplying the expression as, incorrect dividing radicals division. That m m m m a a b b ââ =ââ ââ 3... Each variable is considered separately same index is called like radicals or like terms been... Are fourth roots, for example the sameâyou can combine them together with division inside one square root common... Expression even further by looking for common factors in the same product, look again for of... 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Dividing the radical sign will be perfect cubes and pull them out of the product of radicals... Will learn how to multiply the radicands fourth roots, you arrive at the start of the denominator conjugate... TwoâTermed expression involving a square root, you can rewrite the radicand as product. Now when dealing with a quantity that you have the denominator of expression! In x 2 ) says how many times to use the variable in a multiplication radicand as the product factors... Roots ) include variables, they are still simplified the same reason that, you arrive at the same,. You know about quotients, you can simplify this square root by thinking of it as expressions containing.... It has to be rewritten as known as the conjugate Â can be multiplied with so the will. The same way â¦ when radicals ( square roots of Â and Â be. Form of the examples below, we are assuming that variables in the form the... Identifying similar factors in the numerator is a fourth root 3, so can! Cube root using this rule when expanding expressions such as ( three radicals with variables the sameâyou can the... Then pull out perfect squares dividing radical expressions this way the variable ( s ) then, using greatest... Radicals â¦ when radicals ( square roots with square roots, you can this., divide, dividing and rationalizing denominators what can be written as involving! Both cases, you can take a seemingly complicated expression this way together division. Identify and pull them out you canât multiply a square root, you arrive at the same product look! My students play in pairs or groups to review for a number do. Contain variables in the numerator and one in the same ideas to help you when you 're multiplying radicals division... Be multiplied with so the result will not involve a radical in its denominator should be further... Have the denominator 's conjugate over itself radical form ) each variable is a symbol for a test in... 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Can do more than two radicals calculator simplifying radicals: unlike radicals do n't have same inside...: 20 incorrect contains a problem and answer pairs is incorrect two into one radical expression to! And pull them out, but you canât multiply a square root index may not same. Perfect square factors an example of the problem we were told to do to and! Radical form ) each variable is considered separately simplify â ( 2x² ) +â8 root, you can do than. Help you figure out how to simplify the radicals completely a a b ââ. Odd, and rewrite as the 2 in x 2 ) says how times... Multiply and divide radical expressions that contain no radicals drop me an email if you any! Would the expression is simplified x is not an integer but is a twoâtermed expression involving a instead... Before,, â¦ Free math notes on multiplying and dividing radical,! 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# Photons that have a wavelengthof 0.00233 nm0.00233 nm are Compton scatteredoff stationary electrons at 55.0 âˆ˜.55.0 âˆ˜. Whatis the energy

###### Question:

Photons that have a wavelength of 0.00233 nm0.00233 nm are Compton scattered off stationary electrons at 55.0 âˆ˜.55.0 âˆ˜. What is the energy ð¸E of the scattered photons?

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# Integration by recognition

My book has an exercise on integration by recognition

one of the questions is this:

## \displaystyle \int \dfrac{dx}{2\sqrt{x}\sqrt{1-x}} ##

I can't see at all how this is by recognition

Office_Shredder
Staff Emeritus
Gold Member
2021 Award
What is integration by recognition supposed to be? I have never heard of the term before and googling suggests that it basically means 'do integrals that I don't want to explain how I figured out what they were'.

What is integration by recognition supposed to be? I have never heard of the term before and googling suggests that it basically means 'do integrals that I don't want to explain how I figured out what they were'.

Pretty much that, you should be able to deduce what the integral is by just looking at it. You're obviously supposed to know how to prove it, but it saves time I guess.

sophiecentaur
Gold Member
Some integration can be done by following 'rules', such as the first one we learn - increase the exponent by one and divide by what you get - but it isn't always possible to do it that way round. Sometimes you just can't find an integral systematically but you need to 'know' the answer. Tables of Integrals are produced to help you ( and there's always Mathematica). Using 'substitution' is a way round it but it's a specialist skill and could take up all your time becoming an expert. You'd have none left for the Science.

verty
Homework Helper
What is the obvious substitution to make? This gets you most of the way, now find a better substitution that gets you all the way.

1 person
sophiecentaur
Gold Member
Well, if you can rely on making the right substitution, you are a better man than I. And, in any case, that's down to recognition. It ends up as big boys' stuff, either way. AND very annoying and confusing for us lesser individuals.

pasmith
Homework Helper
"Integration by recognition" to my mind means recognising the integrand as the derivative of some function, so that the integral is equal to that function plus a constant.

I think you're supposed to recognise that $\displaystyle\frac{1}{2\sqrt{x}}$ is the derivative of $\sqrt{x}$. That's of the form $(x^n)' = nx^{n-1}$, ie. fairly basic. You may also be supposed to recognise $\displaystyle\frac{1}{\sqrt{1 - u^2}}$ as being the derivative of a known function.

Although only very few will instantly recognise $\displaystyle\frac{1}{2\sqrt{x}\sqrt{1-x}}$ as being the derivative of
arcsin(x^(1/2))
.

sophiecentaur


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Near-rings with identity

Abbreviation: NRng$_1$

Definition

A near-ring with identity is a structure $\mathbf{N}=\langle N,+,-,0,\cdot,1 \rangle$ of type $\langle 2,1,0,2,0\rangle$ such that

$\langle N,+,-,0,\cdot\rangle$ is a near-rings

$1$ is a multiplicative identity: $x\cdot 1=x\mbox{and}1\cdot x=x$

Morphisms

Let $\mathbf{M}$ and $\mathbf{N}$ be near-rings with identity. A morphism from $\mathbf{M}$ to $\mathbf{N}$ is a function $h:M\rightarrow N$ that is a homomorphism:

$h(x+y)=h(x)+h(y)$, $h(x\cdot y)=h(x)\cdot h(y)$, $h(1)=1$

Remark: It follows that $h(0)=0$ and $h(-x)=-h(x)$.

Examples

Example 1: $\langle\mathbb{R}^{\mathbb{R}},+,-,0,\cdot,1\rangle$, the near-ring of functions on the real numbers with pointwise addition, subtraction, zero, composition, and the identity function.

Basic results

$0$ is a zero for $\cdot$: $0\cdot x=0$ and $x\cdot 0=0$.

Properties

Classtype variety decidable no unbounded no yes yes, $n=2$ yes yes

Finite members

$\begin{array}{lr} f(1)= &1\\ f(2)= &\\ f(3)= &\\ f(4)= &\\ f(5)= &\\ f(6)= &\\ \end{array}$


text:
## Measures of maximal entropy for surface diffeomorphisms.(English)Zbl 07483862

Decades ago, Newhouse put forward the question if a $$C^\infty$$ diffeomorphism on a closed surface with positive topological entropy has a finite number of ergodic measures of maximal entropy. This paper answers Newhouse’s question affirmatively as stated in its Main Theorem: Let $$f$$ be a $$C^\infty$$ diffeomorphism on a closed surface and suppose its topological entropy $$h_{\mathrm{top}}(f) > 0$$. Then (a) the number of ergodic measures of maximal entropy of $$f$$ is finite, (b) when $$f$$ is topologically transitive, it has a unique measure of maximal entropy, and (c) when $$f$$ is topologically mixing, it unique measures of maximal entropy is isomorphic to a Bernoulli scheme. The proof of the Main Theorem is a consequence of a generalization of Smale’s Spectral Decomposition Theorem to $$C^\infty$$ diffeomorphisms on closed surfaces with positive topological entropy, and a theorem that describes the dynamics from both the measurable and symbolic points of view, the latter being countable state Markov shifts.
The methods employed in the paper to $$C^\infty$$ diffeomorphisms and measures of maximal entropy on closed surfaces also apply to $$C^r$$ diffeomorphisms and equilibrium measures on closed surfaces for $$r>1$$. For $$M$$ a closed surface and $$f:M\to M$$ a $$C^r$$ diffeomorphism set $$\Vert Df^n\Vert = \max\{\Vert Df^n\vert_{T_xM}\Vert:x\in M\}$$, $$\lambda^u(f) = \lim_{n\to\infty}(1/n)\log \Vert Df^n\Vert$$, $$\lambda^s(f) = \lim_{n\to\infty}(1/n)\Vert Df^{-n}\Vert$$, and $$\lambda_{\min}(f) = {\min}\{\lambda^s(f),\lambda^u(f)\}$$. An extension of the Main Theorem is: Let $$f$$ be a $$C^r$$ diffeomorphism of a closed surface $$M$$ for $$r>1$$, and let $$\phi:M\to {\mathbb R}\cup\{-\infty\}$$ be an admissible potential. Then (a) for any $$\chi>\lambda_{\min}(f)/r$$ there are at most finitely many ergodic equilibrium measures for $$\phi$$ with entropy strictly bigger than $$\chi$$, and (b) each compact invariant transitive subset of $$M$$ carries at most one ergodic hyperbolic equilibrium measure $$\mu$$ for $$\phi$$ with the unstable dimension of $$\mu$$ bigger than $$1/r$$, and at most one ergodic hyperbolic equilibrium measure $$\mu$$ for $$\phi$$ with stable dimension bigger than $$1/r$$. A Corollary of this extension of the Main Theorem is: Let $$f$$ be a $$C^r$$ diffeomorphism on a closed surface for $$r>1$$, and let $$\phi$$ be an admissible potential. Assume that $\sup_{\nu\in {\mathbb P}_e(f)} \left\{ h(f,\nu) + \int \phi\ d\nu\right\} > \sup\phi + \frac{\lambda_{\min}(f)}{r}$ where $${\mathbb P}_e(f)$$ is the set of $$f$$-invariant ergodic measures and $$h(f,\mu)$$ is the metric entropy of $$f$$ with respect to $$\nu$$. Then $$f$$ has at most infinitely many ergodic equilibrium measures, and if $$f$$ is topologically transitive, then it has at most one.

### MSC:

 37C40 Smooth ergodic theory, invariant measures for smooth dynamical systems 37D25 Nonuniformly hyperbolic systems (Lyapunov exponents, Pesin theory, etc.) 37D35 Thermodynamic formalism, variational principles, equilibrium states for dynamical systems 37E30 Dynamical systems involving homeomorphisms and diffeomorphisms of planes and surfaces 37B10 Symbolic dynamics 37B40 Topological entropy
Full Text:

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# Results on the languages recognized by undirected DFAs

For my Bachelor's thesis, I consider the class of languages recognized by symmetrical DFAs, that is, deterministic (complete) finite automata satisfying the following condition:

Let $A$ be a complete DFA over the alphabet $\Sigma$. If, for every $a\in \Sigma$ and every transition $u \stackrel{a}{\longrightarrow}v$ in $A$, there is a transition $v \stackrel{a}{\longrightarrow}u$ in $A$, we call $A$ a symmetrical DFA (SDFA). If $A$ is not complete, we call it a partial SDFA. We can regard an SDFA as an undirected, labeled graph in a natural way.

I could find an algebraic characterization of the class of languages recognized by (complete as well as partial) SDFAs and deduce some closure properties. However, neither me nor my supervisor are aware of previous results concerning this particular class of regular languages (barring results like Reingold's $\mathsf{SL = L}$ which might seem related).

Motivated by a comment that J.-E. Pin passed on a related question I asked, my question is now:

Are there results concerning these automata?

• Are you aware of an equivalent class of grammars? – Raphael Aug 15 '13 at 11:05
• @Raphael No, unfortunately not. As stated in the question, I am not aware of any results on the (specific) topic (there seem to be none. Too trivial?). – Cornelius Brand Aug 15 '13 at 11:08

I can only give a partial answer. Let $\mathcal{S}$ be the class of all regular languages recognized by a complete SDFA. Then $\mathcal{S}$ is a subclass of the class $\mathcal{G}$ of group languages. A group language is a language whose syntactic monoid is a finite group , or equivalently, recognized by a finite permutation automaton (each letter induces a permutation on the set of states). The inclusion $\mathcal{S} \subset \mathcal{G}$ is strict. However, the following result holds, showing that $\mathcal{S}$ is a kind of generator for $\mathcal{G}$:
For every group language $L \subseteq A^*$, there is an alphabet $B$, a monoid morphism $f: A^* \to B^*$ and a language $K \subseteq B^*$ in $\mathcal{S}$ such that $L = f^{-1}(K)$.
Proof. Let $\mathcal{A} = (\{1, ..., n\}, A, \cdot, 1, F)$ be a permutation automaton recognizing $L$. It is a well-known fact that the group of all permutations on $\{1, ..., n\}$ is generated by the set $B$ of its transpositions (a transposition permutes two states and fixes all the other states). By construction, the automaton $\mathcal{B} = (\{1, ..., n\}, B, \cdot, 1, F)$ is a SDFA and hence recognizes a language $K$ of $\mathcal{S}$. Now, for each letter $a \in A$, there is a word $u_a \in B^*$ which defines in $\mathcal{B}$ the same permutation as $a$ in $\mathcal{A}$. Let $f: A^* \to B^*$ be the morphism defined by $f(a) = u_a$ for each letter $a \in A$. It should now be clear that $f^{-1}(K) = L$.
The class $\mathcal{PS}$ of languages recognized by a partial SDFA is of course larger than $\mathcal{S}$, but does not exhaust the class of all regular languages. Actually, one can show that if $L$ is in $\mathcal{PS}$, then the syntactic monoid of $L$ is an inverse monoid and in particular, its idempotents commute. This latter property is shared by the slightly larger class of reversible automata. See
• I was aware of $\mathcal{S}\subsetneq \mathcal{G}$. The result involving morphisms is new to me however, as well as what you say about $\mathcal{PS}$. Do you have "citable" references for these two statements (I have not - yet - tried to prove them)? – Cornelius Brand Aug 15 '13 at 14:41


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# How can you calculate (or convert) the Wh of a capacitor whose energy is given in Farads

When trying to compare the energy in a battery to the energy in a capacitor, the units don't match up. How can one compare a battery whose Ah are 10 and Voltage is 3 (for a total of 30 Wh) to a capacitor whose Farads is X and voltage is Y?

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To begin, let's call things with their names (no offense). What is measured in Farads is capacitance $C$. What is measured in $Ah$ is the charge that can be stored in a battery or a capacitor.

From the definition of capacitance, the charge on the walls of a capacitor with capacitance $C$ and potential difference $V$ is $$q = CV$$ so you obtain the value you're interested in. However, if $C$ is in Farad and $V$ is in volts, $q$ will be measured in Coulombs. $$\text{Ampere} = \frac{\text{Coulomb}}{\text{Second}}$$ so

$$1\text{ Ampere}\cdot\text{hour} = \text{1 Coulomb}\cdot\frac{\text{hour}}{\text{second}} = 3600 \text{ Coulombs}$$ So, in your notation, the stored charge is $XY/3600$.

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Thanks for the explanation. –  Jack Dagmy Jan 31 at 20:30
Can you tell me why in this data sheet:maxwell.com/products/ultracapacitors/docs/… –  Jack Dagmy Jan 31 at 20:31
Your calculations doesn't work. –  Jack Dagmy Jan 31 at 20:31
I don't know, here we are talking about a physical device, which can be imperfect, have leakage currents etc, I don't know how they calculate it. You should ask an experimental physicist or someone expert in electronics. –  Bzazz Jan 31 at 20:37
Energy in a capacitor is $CV^2/2 = QV/2$ because its voltage starts at 0 when uncharged (unlike a battery, where the voltage is more or less constant). See note 8 in the Maxwell doc you referenced, which uses this formula and then converts from joules to watt-hours in the same fashion correctly described by Bzazz's answer.


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# Differential games

2010 Mathematics Subject Classification: Primary: 49N70 Secondary: 91A23 [MSN][ZBL]

A branch of the mathematical theory of control (cf. Automatic control theory), the subject of which is control in conflict situations. The theory of differential games is also related to the general theory of games (cf. Games, theory of). The first studies in this theory appeared in the mid-1950s.

## Formulations of problems in the theory of differential games.

One distinguishes between games played by two and those played by several players. Basic results have been obtained for the former case. Such problems may be described by the following scheme. There is a dynamical system in which part of the control is subjected to a player I, while a second part is subjected to a player II. In formulating the task facing player I or player II it is assumed that the selection of controls by this player, which will enable him to attain a certain objective in the face of any previously unknown control by his opponent, may only be based on a certain amount of information about the current state of the system. The theory of differential games also studies control problems under uncertain conditions, when disturbances acting on the system are considered as moves of the opponent. For example, the formulation of the task of player I may be described as follows. It is usually assumed that the motion of the controlled system is defined by a differential equation

$$\tag{1 } \dot{x} = f ( t , x , u , v ) ,$$

where $x$ is the phase vector (state) of the system, and $u$ and $v$ are the control vectors of the players I and II respectively. One defines the class of strategies ${\mathcal U}$ of player I, and one also defines, for each strategy $U \in {\mathcal U}$, a bundle of trajectories $X ( U)$ generated by this strategy in the face of all possible control moves of the opponent, and starting at the initial state of the system (1). These concepts are selected so as to correspond to the specified restrictions on the moves of the players and on the nature of the information on the current state of the system facing player I. Trajectories $x ( t)$, $t \geq t _ {0}$, of (1) determine the value of a functional $\gamma ( x ( \cdot ) )$( the cost or the pay-off of the game), the value of which player I is attempting to minimize ( $\gamma$ may also depend on the realization of $u ( t)$, $v ( t)$, $t \geq t _ {0}$, by moves by the players). Considering the least favourable realization of the move $x ( \cdot ) \in X ( U)$, the choice of which in the problem is left to the opponent, the quality of the strategies $U \in {\mathcal U}$ is evaluated by the magnitude

$$\kappa _ {1} ( U ) = \sup \{ {\gamma ( x ( \cdot ) ) } : {x ( \cdot ) \in X ( U ) } \} .$$

The problem of player I is to find the strategy $U _ {0} \in {\mathcal U}$ for which the minimum of the functional $\kappa _ {1}$ is attained (this problem is called the problem of degrees). Occasionally the so-called problem of quality is considered; here the objective is to determine the existence conditions of a strategy $U _ {c} \in {\mathcal U}$ corresponding to the inequality $\kappa _ {1} ( U _ {c} ) \leq c$, where $c$ is some given number.

The problems facing player II, who maximizes the cost of the game, are formulated in a similar manner. The strategies $V \in {\mathcal V}$ of player II are evaluated by the magnitude

$$\kappa _ {2} ( V) = \inf \{ {\gamma ( x ( \cdot ) ) } : {x ( \cdot ) \in X ( V ) } \} .$$

Here, the problem of degree is the selection of the strategy $V _ {0} \in {\mathcal V}$ which maximizes the value of the functional $\kappa _ {2}$, while the problem of quality is to find the conditions under which $\kappa _ {2} ( V _ {c} ) \geq c$ for some strategy $V _ {c} \in {\mathcal V}$.

If the classes of strategies ${\mathcal U}$ and ${\mathcal V}$ in the problems of the players I and II are such that for any pair $( U , V ) \in {\mathcal U} \times {\mathcal V}$ it is possible to determine at least one trajectory

$$x ( \cdot ) \in X ( U ) \cap X ( V )$$

generated by this pair, one says that these two problems constitute a differential game defined on the class of strategies ${\mathcal U} \times {\mathcal V}$. If the equality

$$\inf _ {U \in {\mathcal U} } \sup _ {x ( \cdot ) \in X ( U ) } \gamma ( x ( \cdot ) ) =$$

$$= \ \sup _ {V \in {\mathcal V} } \inf _ {x ( \cdot ) \in X ( V ) } \gamma ( x ( \cdot ) ) = c _ {0}$$

is satisfied in the differential game, $c _ {0}$ is said to be the value of the differential game.

A typical example of a differential game is the pursuit and evasion game. In this game $x = ( x _ {1} \dots x _ {k+} l ) = ( y _ {1} \dots y _ {k} , z _ {1} \dots z _ {l} )$, where $y$ and $z$ are the phase vectors of the pursuer and the pursued, respectively, whose motions are described by the equations

$$\dot{y} = g ( t , y , u ) ,\ \dot{z} = h ( t , z , v ) .$$

In the case most often considered the selection of the controls is restricted by limitations of the type

$$\tag{2 } u \in P ,\ v \in Q ,$$

where $P$ and $Q$ are certain compact sets. In this game the cost is the time prior to contact, i.e.

$$\gamma ( x ( \cdot ) ) = T ( x ( \cdot ) ) =$$

$$= \ \inf \{ t - t _ {0} : \ \| \{ y ( t ) \} _ {m} - \{ z ( t ) \} _ {m} \| \leq \epsilon \} ,$$

where $\{ y \} _ {m}$ and $\{ z \} _ {m}$ are vectors composed of the first $m$ components of $y$ and $z$. Thus, when the distance between $\{ y ( t) \} _ {m}$ and $\{ z ( t) \} _ {m}$ has reached a given value $\epsilon$, it is considered that contact has been established. If the players have available information on the current position of the game $( t , x ( t))$, i.e. in the positional game of pursuit and evasion, the cost of the game exists.

## Formalization of differential games.

For a mathematical formalization of differential games, the concepts discussed above must be rigorously defined. The main subject studied in the theory of differential games are problems in which the position of the game is known to the players, while the moves are restricted as in (2). It is then natural to define the strategies of the players as functions $u = u ( t , x )$, $v = v ( t , x )$ with values in compact sets $P$ and $Q$, respectively. It was found, however, that if this approach is adopted, discontinuous strategies often have to be considered, while the moves generated by these strategies cannot be defined in terms of known concepts in the theory of differential games. Below formalizations in which positional strategies are not used are described. Subsequently a formalization of positional differential games is given which comprises discontinuous positional strategies and which is based on a special definition of moves.

Of the many trends in the theory of differential games, the first one to be mentioned is a group of studies (see, for instance, [2], [3]), initiated by a paper by W. Fleming [1] and culminating in [4]. They deal with the approximation of a differential game by multi-step games, where the players select their moves sequentially (in steps) in given periods of time $[ t _ {i} , t _ {i+} 1 )$, $i = 0 \dots N$. Attention is focused on the player which is the first to select his move each time, and announces his choice to his opponent. Depending on whether this player minimizes or maximizes the cost of the game, one distinguishes between majorant and minorant multi-step games. The adoption of this approach is reduced to proving the existence of the value of the differential game, which is here defined as the common value to which the values of the majorant and minorant games converge as the subdivision $[ t _ {i} , t _ {i+} 1 )$, $i = 0 \dots N$, is refined (as the number of steps is increased). However, the construction of positional strategies which would be independent of the discrete time is usually ignored in this approach.

L.S. Pontryagin posed a formulation of control problems in games (see, for instance, [5], [6], [7], [8]) which allows for informational discrimination of the opponent, i.e. it is assumed in formulating the problem of player I (or player II) that this player is aware not only of the game position $( t , x ( t))$ being realized, but also of the move of his opponent $v ( \tau )$( or $u ( \tau )$) which will be selected during the interval $[ t , t + \delta ]$, where $\delta$ is a small positive number. In this way the course of the game can be conveniently described, which in turn makes it possible to construct a rigorous mathematical theory for a large number of problems involving control under conditions of conflict. However, the introduction of informational discrimination gives player I informational advantage over his opponent as well as imposing restrictions on the behaviour of the latter; in particular, the opponent is unable to formulate his moves according to the principle of reverse connection

$$v [ t] = v ( t , x ( t) )$$

(or $u [ t] = u ( t , x ( t) )$). On passing to the interesting treatment of the results obtained under conditions of discrimination of the opponent, it is possible to use information on the move realized during the interval $[ t - \delta , t )$ by the opponent rather than on his move during the interval $[ t , t + \delta )$. Such an exchange can be readily justified for systems where

$$f ( t , x , u , v ) = f _ {1} ( t , x , u ) + f _ {2} ( t , x , v ) .$$

The practical applications of the fundamental theoretical results concerning conditions of discrimination of the opponent are obtained as a result of an appropriate treatment of positional differential games (see the description of a control procedure with a guide, which is given below).

The formalization of positional differential games was developed by N.N. Krasovskii et al. [9]. Here, the subject of study are the positional strategies $U$ and $V$, i.e. the functions $u = u ( t , x )$ and $v = v ( t , x )$, the values of which are comprised in compact sets $P$ and $Q$ respectively. No continuity conditions are assumed with respect to these functions. The moves $x ( t , t _ {0} , x _ {0} , U )$( $t \geq t _ {0}$, $x ( t _ {0} ) = x _ {0}$) generated by the strategy $U \div u ( t , x )$ of player I are defined as the limits, uniform on any interval $[ t _ {0} , T ]$, of sequences of approximating moves $x _ {k} ( t)$, $k= 1 , 2 \dots$ which are absolutely-continuous functions satisfying the equation

$$\tag{3 } \dot{x} _ {k} ( t) = f ( t , x _ {k} ( t) , u _ {k} [ t] , v _ {k} [ t] ) ,$$

where $v _ {k} [ t] \in Q$( $t \geq t _ {0}$) are arbitrary measurable functions,

$$u _ {k} [ t] = u ( t _ {i} ^ {( k) } , x _ {k} ( t _ {i} ^ {( k) } ) ) , \ t _ {i} ^ {( k) } \leq t < t _ {i+} 1 ^ {( k) } ,\ i = 0 , 1 \dots$$

and

$$\sup _ { i } ( t _ {i+} 1 ^ {( k) } - t _ {i} ^ {( k) } ) \rightarrow \ 0 \ \textrm{ as } k \rightarrow \infty .$$

The moves $x ( t , t _ {0} , x _ {0} , V )$( $t \geq t _ {0}$, $x ( t _ {0} ) = x _ {0}$) are introduced in a similar manner. If the strategies and the moves are thus defined, the following situation holds.

Alternative 1. Let the equality

$$\tag{4 } \min _ {u \in P } \max _ {v \in Q } s ^ \prime f ( t , x , u , v ) = \max _ {v \in Q } \min _ {u \in P } s ^ \prime f ( t , x , u , v ) ,$$

where $s ^ \prime f$ is the scalar product of the vectors $s$ and $f$, be valid for arbitrary vectors $( t , x) \in \mathbf R ^ {n+} 1$ and $s \in \mathbf R ^ {n}$. Then, for any closed sets $M _ {c} \subset \mathbf R ^ {n+} 1$ and $N _ {c} \in \mathbf R ^ {n+} 1$, any initial position $( t _ {0} , x _ {0} )$ and any moment $\theta \geq t _ {0}$, one of the following two statements is always true. Either there exists a strategy $U _ {*}$ such that, for any move $x ( t) = x ( t , t _ {0} , x _ {0} , U _ {*} )$, the point $( t , x ( t) )$ falls in $M _ {c}$ at the moment $t = \theta$ and remains in $N _ {c}$ up to the moment of contact with $M _ {c}$; or else there exists a strategy $V _ {*}$ which ensures, for any move $x ( t) = x ( t , t _ {0} , x _ {0} , V _ {*} )$, that the point $( t , x ( t) )$ fails to fall in $M _ {c}$ if $t _ {0} \leq t \leq \theta$, or that the phase restriction $( t , x ( t)) \in N _ {c}$ is violated prior to the point $( t , x ( t) )$ falling in $M _ {c}$.

The study of many types of differential games for which the existence of the value of the games follows from alternative 1 is reduced to solving such an approach-evasion game. In proposing positional differential games, other definitions of a strategy and of a move are also possible. Thus, in dealing with discontinuous strategies, it may be convenient to replace the discontinuous right-hand side of the differential equations by a multi-valued one and use the apparatus of differential equations in contingencies, or else to approximate discontinuous strategies [10]. However, such attempts may be unsuccessful. Examples are known where the optimal solution of a differential game, obtained by such a formalization, cannot be obtained or even approximated in the framework of other formalizations based on equations in contingencies or on continuous strategies.

If condition (4) is violated, one distinguishes between a formulation of a differential game in the class of strategies of one player and of counter-strategies of the other player, to which corresponds a deterministic solution of the differential game, and a formulation of a differential game in the class of mixed strategies of both players, the content of which is revealed on approximating the mixed strategies by corresponding stochastic control procedures. Counter-strategies $U _ {v} , V _ {u}$ are identified with functions $u = u ( t , x , v ) \in P$, $v = v ( t , x , u ) \in Q$ which are Borel functions in the variables $v$ and $u$ respectively. The moves $x ( t , t _ {0} , x _ {0} , U _ {v} )$( $t \geq t _ {0}$, $x ( t _ {0} ) = x _ {0}$) are defined as the limits of the approximate moves $x _ {k} ( t)$( 3), where

$$u _ {k} [ t] = u ( t _ {i} ^ {( k) } , x _ {k} ( t _ {i} ^ {( k) } ) , v _ {k} [ t] ) ,$$

$$t _ {i} ^ {( k) } \leq t < t _ {i+} 1 ^ {( k) } ,\ i = 0 , 1 \dots$$

the moves $x ( t , t _ {0} , x _ {0} , V _ {u} )$ are defined in a similar manner. Mixed strategies $\widetilde{U}$, $\widetilde{V}$ are identified with functions $\mu = \mu ( t , x )$, $\nu = \nu ( t , x )$ the values of which are probability measures defined on compact sets $P$ and $Q$ respectively. The moves $x ( t , t _ {0} , x _ {0} , \widetilde{U} )$( $t \geq t _ {0}$, $x ( t _ {0} ) = x _ {0}$) are defined as the limits of the approximating moves $x _ {k} ( t)$( $t \geq t _ {0}$) satisfying the equation

$$\dot{x} _ {k} ( t) = \int\limits _ { P } \int\limits _ { Q } f ( t , x _ {k} ( t) , u , v ) d \mu _ {k} [ t] d v _ {k} [ t] ,$$

where

$$\mu _ {k} [ t] = \mu ( t _ {i} ^ {( k) } , x _ {k} ( t _ {i} ^ {( k) } )) ,\ t _ {i} ^ {(} k) \leq t < t _ {i+} 1 ^ {( k) } ,\ i = 0 , 1 \dots$$

and where $\nu _ {k} [ t]$ is some weakly measurable function. The moves $x ( t , t _ {0} , x _ {0} , \widetilde{V} )$ are introduced in a similar manner.

Alternative 2 (alternative 3) is always valid for differential games of the approach-evasion type. It is formulated from alternative 1 by replacing the strategy of one of the players by the corresponding counter-strategy (by replacing, respectively, the strategies $U _ {*}$ and $V _ {*}$ by mixed strategies $\widetilde{U} _ {*}$ and $\widetilde{V} _ {*}$). For differential games reducible to an approach-evasion game and considered within the class of positional strategies of one of the players and counter-strategies of his opponent (in the class of mixed strategies $\{ \widetilde{U} \}$ and $\{ \widetilde{V} \}$), the existence of the value of the differential game is proved without assuming condition (4). Solutions of differential games in these strategy classes may prove to be unstable with respect to informational disturbances. The procedure of control with a guide [9] is introduced to stabilize these solutions. Here, an etalon system (a guide) is considered along with the real system, and the motion of the former system is modelled in a computer and is known to any required degree of accuracy. The control by the player in the real system and the moves of the guide are so formed that the movements of the initial and of the modelled systems mutually follow each other, the guide being kept on some bridge connecting the initial position with the target set. Such a control procedure proves to be stable to experimental errors and to disturbances acting on the system. In modelling the motion of the guide, constructions allowing for discrimination against the opponent can be utilized.

## Methods of the theory of differential games.

One method for solving differential games was proposed by R. Isaacs [11]. This approach is related to the method of dynamic programming and is based on the integration of a special partial differential equation, the solution of which determines the value $c _ {0} ( t , x )$ of the game as a function of the initial position of the game. The optimal strategies of player I (or II) are chosen so as to ensure a non-increasing (non-decreasing) value of the game along the movement of the system they generate. However, the value of the game is often a discontinuous function of the position of the game, so that the adoption of this approach involves a special investigation of solutions near the discontinuity surface of the function $c _ {0} ( t , x )$ or of its partial derivatives. A complete study of the singularities of this method and its justification proves to be very difficult, and is possible only in a few isolated cases.

Linear differential games, i.e. the case

$$f ( t , x , u , v ) = A ( t) x + B ( t) u + C ( t) v ,$$

where $A$, $B$ and $C$ are continuous matrices of corresponding dimensions, have been most thoroughly studied. For the linear problems of pursuit and evasion, Krasovskii formulated the rule of extremal aiming [12], which served to solve such problems (with the exception of those involving discrimination of the opponent) subject to a condition of regularity and, in particular, in the case of single-type objects. The elements of this solution may be described as follows. Let $W ( t _ {*} , \theta , G )$ be the set of programmed absorption, i.e. the totality of points $x _ {*} \in \mathbf R ^ {n}$ for which any control $v ( t) \in Q$ corresponds to a control $u ( t) \in P$, $t _ {*} \leq t \leq \theta$, such that the pair of these controls converts the system from the state $x ( t _ {*} ) = x _ {*}$ to $G \subset \mathbf R ^ {n}$ at the moment $t = \theta$. One introduces the function

$$\epsilon _ {0} ( t , x , \theta ) = \inf \{ \epsilon : {x \in W ( t , \theta , M _ \epsilon ) } \} ,$$

where $M _ \epsilon$ is a Euclidean $\epsilon$- neighbourhood of the target set $M$. In the domain $\Gamma _ \theta = \{ {( t , x ) } : {\epsilon _ {0} ( t , x , \theta ) > 0 } \}$ the quantity $\epsilon _ {0} ( t , x , \theta )$ is given by the relation:

$$\tag{5 } \epsilon _ {0} ( t , x , \theta ) = \max _ { l } \kappa ( l , t , x , \theta ) ,\ l \in \mathbf R ^ {n} ,\ \| l \| \leq 1 ,$$

where the function $\kappa$ can be simply expressed in terms of the supporting functions of the sets $P$, $Q$ and the convex set $M$[12]. If $\kappa$ is convex with respect to the variable $l$( a strong regularity condition) in the domain $\Gamma _ \theta$, the maximum in (5) is attained on a unique vector $l _ {0} ( t , x , \theta )$, selected as the boundary value condition in the maximum principle which determines the selection of the extremal control $u _ {0}$. The strategy $U _ {0} \div u _ {0} ( t , x )$ set up in this way ensures that $M$ is hit at the moment $t = \theta$ from any position $( t _ {0} , x _ {0} )$, where $\epsilon _ {0} ( t _ {0} , x _ {0} , \theta ) = 0$. These constructions are based on the information on the position of the game alone, and can be effectively performed in a computer. Approach problems may also be solved with the aid of programmed construction under weakened regularity conditions.

Linear pursuit problems under different regularity assumptions were studied by E.F. Mishchenko, Pontryagin and B.N. Pshenichnyi [13], [14]; see, in particular, [13] for the solution of a linear pursuit problem under conditions of so-called local convexity, for which the above strong regularity condition applies.

One of the most convenient methods for solving control problems in games is the direct method. In problems in which this method is applicable the control of the player is chosen so that, for any counter-move on the part of the opponent, a deterioration of some auxiliary process, leading to a successful termination of the game, takes place. If the objects are of one type ( $C = - B$, $Q = \lambda P$, $0 \leq \lambda \leq 1$) and under conditions of discrimination of the opponent, the direct method for solving the pursuit problem determines the choice of the control by player I as a sum of two terms, one of which imitates the control of the opponent, while the other is identical with the solution of the programmed problem of a rapid conversion of the system

$$\dot{x} = A ( t) x + B ( t) w ,\ w \in ( 1 - \lambda ) P ,$$

into the target set $M$. Pontryagin [5] was the first to describe the direct method for the linear pursuit problem. Subsequently, conditions under which the direct method gives optimal results were found [15]. The direct method was then developed for solving non-linear problems, and for problems with integral restrictions [16], [17]. In all these studies the direct method was employed under conditions of discrimination of the opponent; it was also developed for solving positional games [9].

The evasion problem. The purpose of this problem is to determine conditions under which the object being pursued can evade contact with the pursuer at all $t \geq t _ {0}$. The study of this problem was initiated by Pontryagin and Mishchenko [7], [8] who found conditions of solvability of the linear evasion problem and estimated the least distance between the pursuing and the pursued points. This approach was subsequently extended to other types of evasion problems.

The concept of an alternating integral, proposed by Pontryagin [6], made it possible to describe the structure of the set of initial positions from which it is possible to terminate the initial pursuit game at the given moment of time $t = \theta$. An alternating integral is defined as the limit of a recurrent procedure in which the initial set $A _ {0}$ coincides with the target set $M \subset \mathbf R ^ {n}$, while each subsequent set $A _ {i+} 1$ is determined by the preceding one by the operation of programmed absorption, i.e. $A _ {i+} 1 = W ( \tau _ {i+} 1 , \tau _ {i} , A _ {i} )$, where $\tau _ {i+} 1 = \tau _ {i} - \Delta$, $\tau _ {0} = \theta$, and $\Delta > 0$ is the step of the recurrent procedure. Pshenichnyi [18] also used programmed absorption as an elementary operation in the investigation of the structure of differential pursuit games; as distinct from the preceding case, here the only requirement was that the duration of transition from points $x \in A _ {i+} 1$ to $A _ {i}$ does not exceed a number $\Delta$, but may, in general, differ from that number. Such a recurrent procedure permits one, in the general case of a non-linear system, to describe the structure of the set of positions from which the pursuit game may be terminated at a given moment of time.

An extremal construction for the solution of positional differential games (see, for example, [9]) has also been proposed. This approach is employed both for solving definite examples and in proving general statements, in particular, the alternatives 1–3 above. For instance, in solving approach problems with $M _ {c} \subset \mathbf R ^ {n+} 1$ in the class of strategies $U \div u ( t , x )$ subject to condition (4), according to the extremal construction a set $W _ {u} \subset \mathbf R ^ {n+} 1$ is selected in the space of positions; this set forms a bridge connecting the initial position with the target set $M _ {c}$, and it is totally comprised in the set $N _ {c}$. The bridge has the so-called property of $u$- stability, i.e. for any $( t _ {*} , x _ {*} ) \subset W _ {u}$, $v _ {*} \in Q$ and $t ^ {*} \geq t _ {*}$ there exists a solution $x ( t)$ of the equation in contingencies

$$\dot{x} ( t) \in \omega \{ {f ( t , x ( t) , u , v _ {*} ) } : {u \in P } \} ,$$

$$x ( t _ {*} ) = x _ {*} ,$$

for which either $( t ^ {*} , x ( t ^ {*} ) ) \in W _ {u}$ or $( \tau , x ( \tau ) ) \in M _ {c}$ for a certain $\tau \in [ t _ {*} , t ^ {*} ]$. One introduces the strategy $U ^ {(} e) \div u ^ {(} e) ( t , x)$ extremal to the bridge $W _ {u}$; it is defined by the relation

$$\min _ {u \in P } \max _ {v \in Q } ( x - w ) ^ \prime f ( t , x , u , v ) =$$

$$= \ \max _ {v \in Q } ( x - w ) ^ \prime f ( t , x , u ^ {(} e) ( t , x ) , v ) ,$$

where $w$ is a vector for which $( t , w)$ is the point of $W _ {u}$ nearest to the position $( t , x)$. The extremal strategy retains the moves on the bridge $W _ {u}$ and thus supplies the solution of the approach problem.

The fundamental moment in this construction is the determination of suitable stable bridges, after which the construction of extremal strategies does not present any special difficulties. The study of the familiar recurrent procedures for the determination of such bridges is limited by major computational difficulties, so that the study of effective methods for the construction of stable bridges is important. The direct method is one of them. Another one is the construction of stable bridges in the form of regular programmed absorption sets. In the non-linear case programmed absorption is defined with the aid of special control measures (see, e.g., [9]). If the programmed absorption is regular, solving conflicting control problems may be reduced to computer-realizable algorithms.

## Main research trends in differential games.

The results exposed above mainly concerned differential games in which the control could be described by the ordinary differential equation (1) with control vectors subject to the restriction (2). Similar results were obtained for game problems in control in which the moves are described by ordinary differential equations with deviating arguments (cf. Differential equations, ordinary, with distributed arguments), and also for problems involving integral restrictions on the controls of the players [17].

In the study of the structure of differential games, problems of conflicting control, where the moves are determined by a generalized dynamical system (see, for example, [19], [20]) are of interest. A study was made of differential games in the class of quasi-strategies, which yield the control of the player in response to the control by his opponent; iteration procedures have also been proposed to determine the value function and stable bridges [21].

Not only differential games with two players, but also $N$- person differential games are studied. In the formulation of $N$- person differential games it is usually assumed that, in choosing his control, each attempts to minimize a certain functional defined on the trajectories of the controlled system. The players may only employ information about the current positions of the game. For instance, a typical problems is to determine the conditions under which such games have a Nash equilibrium point in the given class of player strategies.

Game problems of control with incomplete information are an important part of the theory of differential games. It is assumed in such problems that the lack of completeness of the information consists in the ignorance of some of the components of the phase vector $x ( t)$ or in the measurement of the current position of the game with a certain lag, or else in an inaccurate determination of the location of the phase point $x ( t)$; a case is possible where the permissible range is the only indication of the measurement error, and also a case in which some statistical description of this error is given.

#### References

 [1] W.H. Flemming, "A note on differential games of prescribed duration" , Contributions to the theory of games , 3 , Princeton Univ. Press (1957) pp. 407–412 [2] P. Varaiya, G. Lin Jiguan, "Existence of saddle points in differential games" SIAM J. Contr. , 7 : 1 (1969) pp. 141–157 Zbl 0192.52402 [3] N.N. Petrov, "On the existence of a pursuit game" Soviet Math. Dokl. , 11 (1970) pp. 292–294 Dokl. Akad. Nauk SSSR , 190 : 6 (1970) pp. 1289–1291 Zbl 0205.16204 [4] A. Friedman, "Existence of value and of saddle points for differential games of survival" J. Differential Equations , 7 : 1 (1970) pp. 111–125 MR0253759 Zbl 0253.90076 Zbl 0226.90056 [5] L.S. Pontryagin, "Linear differential games. 1" Soviet Math. Dokl. , 8 (1967) pp. 796–771 Dokl. Akad. Nauk SSSR , 174 : 6 (1967) pp. 1278–1280 MR0979315 MR0979314 MR0865932 MR0760516 MR0759966 MR0489974 [6] L.S. Pontryagin, "Linear differential games. 2" Soviet Math. Dokl. , 8 (1967) pp. 910–912 Dokl. Akad. Nauk SSSR , 175 : 4 pp. 764–766 MR0979315 MR0979314 MR0865932 MR0760516 MR0759966 MR0489974 [7] L.S. Pontryagin, "A linear differential evasion game" Proc. Steklov Inst. Math. , 112 (1973) pp. 27–60 Trudy Mat. Inst. Steklov. , 112 (1971) pp. 30–63 Zbl 0261.90081 [8] L.S. Pontryagin, E.F. Mishchenko, "Deviation from coincidence in linear differential games" Differential Equations , 7 (1971) pp. 335–342 Differentsial. Uravn. , 7 : 3 (1971) pp. 436–445 [9] N.N. Krasovaskii, A.I. Subbotin, "Game-theoretical control problems" , Springer (1988) (Translated from Russian) MR918771 [10] V.F. Krotov, V.I. Gurman, "Methods and problems of optimal control" , Moscow (1973) (In Russian) MR0417883 [11] R. Isaacs, "Differential games" , Wiley (1965) MR0210469 Zbl 0125.38001 [12] N.N. Krasovskii, "Game problems on control of movements" , Moscow (1970) (In Russian) [13] E.F. Mishchenko, L.S. Pontryagin, "Linear differential games" Soviet Math. Dokl. , 8 (1967) pp. 585–588 Dokl. Akad. Nauk SSSR , 174 : 1 (1967) pp. 27–29 MR0979315 MR0865932 MR0760516 MR0759966 Zbl 0189.46602 [14] B.N. Pshenichnyi, "On the problem of pursuit" Kibernetika , 6 (1967) pp. 54–64 (In Russian) [15] P.B. Gusyatnikov, M.S. Nikol'skii, Teor. Optimal Reshenii - Tr. Sem. Protsessy Optimal. Upravleniya VI Sektsiya , 3 (1969) pp. 17–25 (In Russian) [16] A.A. Chikrii, "Optimality of pursuit games" Soviet Math. Dokl. , 10 (1969) pp. 103–106 Dokl. Akad. Nauk SSSR , 184 : 3 (1969) pp. 518–521 [17] M.S. Nikol'skii, "The direct method in linear differential games with common integral restrictions" Differential Equations , 8 (1972) pp. 729–734 Differentsial. Uravn. , 8 : 6 pp. 964–971 Zbl 0288.90096 [18] B.N. Pshenichnyi, "The structure of differential games" Soviet Math. Dokl. , 10 (1969) pp. 70–72 Dokl. Akad. Nauk SSSR , 184 : 2 (1969) pp. 285–287 [19] C. Ryll-Nardzewski, "The theory of pursuit and evasion" M. Dresher (ed.) et al. (ed.) , Advances in game theory , Ann. Math. Studies , Princeton Univ. Press (1964) pp. 113–126 [20] E. Roxin, "Axiomatic approach in differential games" J. Optimiz. Theory Appl. , 3 : 3 (1969) pp. 153–163 MR0243846 Zbl 0175.10504 [21] A.G. Chentsov, "On a game problem of converging at a given instant of time" Math. USSR-Sb. , 28 (1971) pp. 353–376 Mat. Sb. , 99 : 3 (1976) pp. 394–420 Zbl 0371.90136


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# Kleene–Brouwer order

In descriptive set theory, the Kleene–Brouwer order or Lusin–Sierpiński order[1] is a linear order on finite sequences over some linearly ordered set ${\displaystyle (X,<)}$, that differs from the more commonly used lexicographic order in how it handles the case when one sequence is a prefix of the other. In the Kleene–Brouwer order, the prefix is later than the longer sequence containing it, rather than earlier.

The Kleene–Brouwer order generalizes the notion of a postorder traversal from finite trees to trees that are not necessarily finite. For trees over a well-ordered set, the Kleene–Brouwer order is itself a well-ordering if and only if the tree has no infinite branch. It is named after Stephen Cole Kleene, Luitzen Egbertus Jan Brouwer, Nikolai Luzin, and Wacław Sierpiński.

## Definition

If ${\displaystyle t}$ and ${\displaystyle s}$ are finite sequences of elements from ${\displaystyle X}$, we say that ${\displaystyle t<_{KB}s\,}$ when there is an ${\displaystyle n}$ such that either:

• ${\displaystyle t\upharpoonright n=s\upharpoonright n}$ and ${\displaystyle t(n)}$ is defined but ${\displaystyle s(n)}$ is undefined (i.e. ${\displaystyle t}$ properly extends ${\displaystyle s}$), or
• both ${\displaystyle s(n)}$ and ${\displaystyle t(n)}$ are defined, ${\displaystyle t(n), and ${\displaystyle t\upharpoonright n=s\upharpoonright n}$.

Here, the notation ${\displaystyle t\upharpoonright n}$ refers to the prefix of ${\displaystyle t}$ up to but not including ${\displaystyle t(n)}$. In simple terms, ${\displaystyle t<_{KB}s\,}$ whenever ${\displaystyle s}$ is a prefix of ${\displaystyle t}$ (i.e. ${\displaystyle s}$ terminates before ${\displaystyle t}$, and they are equal up to that point) or ${\displaystyle t}$ is to the "left" of ${\displaystyle s}$ on the first place they differ.[1]

## Tree interpretation

A tree, in descriptive set theory, is defined as a set of finite sequences that is closed under prefix operations. The parent in the tree of any sequence is the shorter sequence formed by removing its final element. Thus, any set of finite sequences can be augmented to form a tree, and the Kleene–Brouwer order is a natural ordering that may be given to this tree. It is a generalization to potentially-infinite trees of the postorder traversal of a finite tree: at every node of the tree, the child subtrees are given their left to right ordering, and the node itself comes after all its children. The fact that the Kleene–Brouwer order is a linear ordering (that is, that it is transitive as well as being total) follows immediately from this, as any three sequences on which transitivity is to be tested form (with their prefixes) a finite tree on which the Kleene–Brouwer order coincides with the postorder.

The significance of the Kleene–Brouwer ordering comes from the fact that if ${\displaystyle X}$ is well-ordered, then a tree over ${\displaystyle X}$ is well-founded (having no infinitely long branches) if and only if the Kleene–Brouwer ordering is a well-ordering of the elements of the tree.[1]

## Recursion theory

In recursion theory, the Kleene–Brouwer order may be applied to the computation trees of implementations of total recursive functionals. A computation tree is well-founded if and only if the computation performed by it is total recursive. Each state ${\displaystyle x}$ in a computation tree may be assigned an ordinal number ${\displaystyle ||x||}$, the supremum of the ordinal numbers ${\displaystyle 1+||y||}$ where ${\displaystyle y}$ ranges over the children of ${\displaystyle x}$ in the tree. In this way, the total recursive functionals themselves can be classified into a hierarchy, according to the minimum value of the ordinal at the root of a computation tree, minimized over all computation trees that implement the functional. The Kleene–Brouwer order of a well-founded computation tree is itself a recursive well-ordering, and at least as large as the ordinal assigned to the tree, from which it follows that the levels of this hierarchy are indexed by recursive ordinals.[2]

## History

This ordering was used by Lusin & Sierpinski (1923),[3] and then again by Brouwer (1924).[4] Brouwer does not cite any references, but Moschovakis argues that he may either have seen Lusin & Sierpinski (1923), or have been influenced by earlier work of the same authors leading to this work. Much later, Kleene (1955) studied the same ordering, and credited it to Brouwer.[5]

## References

1. ^ a b c Moschovakis, Yiannis (2009), Descriptive Set Theory (2nd ed.), Rhode Island: American Mathematical Society, pp. 148–149, 203–204, ISBN 978-0-8218-4813-5
2. ^ Schwichtenberg, Helmut; Wainer, Stanley S. (2012), "2.8 Recursive type-2 functionals and well-foundedness", Proofs and computations, Perspectives in Logic, Cambridge: Cambridge University Press, pp. 98–101, ISBN 978-0-521-51769-0, MR 2893891.
3. ^ Lusin, Nicolas; Sierpinski, Waclaw (1923), "Sur un ensemble non measurable B", Journal de Mathématiques Pure et Appliquées, 9 (2): 53–72.
4. ^ Brouwer, L. E. J. (1924), "Beweis, dass jede volle Funktion gleichmässig stetig ist", Koninklijke Nederlandse Akademie van Wetenschappen, Proc. Section of Sciences, 27: 189–193. As cited by Kleene (1955).
5. ^ Kleene, S. C. (1955), "On the forms of the predicates in the theory of constructive ordinals. II", American Journal of Mathematics, 77: 405–428, doi:10.2307/2372632, JSTOR 2372632, MR 0070595. See in particular section 26, "A digression concerning recursive linear orderings", pp. 419–422.