In these problems one cannot take as approximate solutions the elements of minimizing sequences. adjective badly or inadequately defined; vague: He confuses the reader with ill-defined terms and concepts. Now I realize that "dots" does not really mean anything here. Experiences using this particular assignment will be discussed, as well as general approaches to identifying ill-defined problems and integrating them into a CS1 course. E. C. Gottschalk, Jr., Jr., Jr., Jr., Jr., Jr., Jr., Jr., Jr., Jr., Jr., Jr., Jr., Jr., Jr., Jr., Jr., Jr., Jr., Jr., Jr., Jr., Jr., Jr., Jr., Jr., Jr., Jr., Jr., Jr., Jr., Jr., Jr., Jr., Jr. What is a post and lintel system of construction what problem can occur with a post and lintel system provide an example of an ancient structure that used a post and lintel system? equivalence classes) are written down via some representation, like "1" referring to the multiplicative identity, or possibly "0.999" referring to the multiplicative identity, or "3 mod 4" referring to "{3 mod 4, 7 mod 4, }". Gestalt psychologists find it is important to think of problems as a whole. c: not being in good health. ill-defined. Meaning of ill in English ill adjective uk / l / us / l / ill adjective (NOT WELL) A2 [ not usually before noun ] not feeling well, or suffering from a disease: I felt ill so I went home. Furthermore, competing factors may suggest several approaches to the problem, requiring careful analysis to determine the best approach. Department of Math and Computer Science, Creighton University, Omaha, NE. Ill defined Crossword Clue The Crossword Solver found 30 answers to "Ill defined", 4 letters crossword clue. After stating this kind of definition we have to be sure that there exist an object with such properties and that the object is unique (or unique up to some isomorphism, see tensor product, free group, product topology). Document the agreement(s). Now, how the term/s is/are used in maths is a . In mathematics, a well-defined expressionor unambiguous expressionis an expressionwhose definition assigns it a unique interpretation or value. An expression is said to be ambiguous (or poorly defined) if its definition does not assign it a unique interpretation or value. Synonyms: unclear, vague, indistinct, blurred More Synonyms of ill-defined Collins COBUILD Advanced Learner's Dictionary. The so-called smoothing functional $M^\alpha[z,u_\delta]$ can be introduced formally, without connecting it with a conditional extremum problem for the functional $\Omega[z]$, and for an element $z_\alpha$ minimizing it sought on the set $F_{1,\delta}$. 2. a: causing suffering or distress. The answer to both questions is no; the usage of dots is simply for notational purposes; that is, you cannot use dots to define the set of natural numbers, but rather to represent that set after you have proved it exists, and it is clear to the reader what are the elements omitted by the dots. Under the terms of the licence agreement, an individual user may print out a PDF of a single entry from a reference work in OR for personal use (for details see Privacy Policy and Legal Notice). Here are the possible solutions for "Ill-defined" clue. \label{eq1} Equivalence of the original variational problem with that of finding the minimum of $M^\alpha[z,u_\delta]$ holds, for example, for linear operators $A$. $$. In the second type of problems one has to find elements $z$ on which the minimum of $f[z]$ is attained. This poses the problem of finding the regularization parameter $\alpha$ as a function of $\delta$, $\alpha = \alpha(\delta)$, such that the operator $R_2(u,\alpha(\delta))$ determining the element $z_\alpha = R_2(u_\delta,\alpha(\delta)) $ is regularizing for \ref{eq1}. - Provides technical . We will try to find the right answer to this particular crossword clue. Does ZnSO4 + H2 at high pressure reverses to Zn + H2SO4? @Arthur So could you write an answer about it? Here are a few key points to consider when writing a problem statement: First, write out your vision. To test the relation between episodic memory and problem solving, we examined the ability of individuals with single domain amnestic mild cognitive impairment (aMCI), a . $$. Make it clear what the issue is. M^\alpha[z,u_\delta,A_h] = \rho_U^2(A_hz,u_\delta) + \alpha\Omega[z], poorly stated or described; "he confuses the reader with ill-defined terms and concepts". \bar x = \bar y \text{ (In $\mathbb Z_8$) } Numerical methods for solving ill-posed problems. The school setting central to this case study was a suburban public middle school that had sustained an integrated STEM program for a period of over 5 years. Is it possible to rotate a window 90 degrees if it has the same length and width? Suppose that in a mathematical model for some physical experiments the object to be studied (the phenomenon) is characterized by an element $z$ (a function, a vector) belonging to a set $Z$ of possible solutions in a metric space $\hat{Z}$. Abstract algebra is another instance where ill-defined objects arise: if $H$ is a subgroup of a group $(G,*)$, you may want to define an operation Therefore this definition is well-defined, i.e., does not depend on a particular choice of circle. Beck, B. Blackwell, C.R. How can I say the phrase "only finitely many. To save this word, you'll need to log in. Evidently, $z_T = A^{-1}u_T$, where $A^{-1}$ is the operator inverse to $A$. This article was adapted from an original article by V.Ya. The best answers are voted up and rise to the top, Not the answer you're looking for? (c) Copyright Oxford University Press, 2023. At first glance, this looks kind of ridiculous because we think of $x=y$ as meaning $x$ and $y$ are exactly the same thing, but that is not really how $=$ is used. $$ Browse other questions tagged, Start here for a quick overview of the site, Detailed answers to any questions you might have, Discuss the workings and policies of this site. Ill-Defined The term "ill-defined" is also used informally to mean ambiguous . Otherwise, the expression is said to be not well defined, ill defined or ambiguous. The existence of quasi-solutions is guaranteed only when the set $M$ of possible solutions is compact. We can reason that $g\left(\dfrac mn \right) = \sqrt[n]{(-1)^m}$ Spline). Is this the true reason why $w$ is ill-defined? Inom matematiken innebr vldefinierad att definitionen av ett uttryck har en unik tolkning eller ger endast ett vrde. Its also known as a well-organized problem. Figure 3.6 shows the three conditions that make up Kirchoffs three laws for creating, Copyright 2023 TipsFolder.com | Powered by Astra WordPress Theme. $h:\mathbb Z_8 \to \mathbb Z_{12}$ defined by $h(\bar x) = \overline{3x}$. Is a PhD visitor considered as a visiting scholar? If you know easier example of this kind, please write in comment. (mathematics) grammar. $$ SIGCSE Bulletin 29(4), 22-23. In the smoothing functional one can take for $\Omega[z]$ the functional $\Omega[z] = \norm{z}^2$. The regularization method. The following are some of the subfields of topology. Allyn & Bacon, Needham Heights, MA. If the conditions don't hold, $f$ is not somehow "less well defined", it is not defined at all. It is critical to understand the vision in order to decide what needs to be done when solving the problem. Document the agreement(s). worse wrs ; worst wrst . Spangdahlem Air Base, Germany. Learn more about Stack Overflow the company, and our products. In formal language, this can be translated as: $$\exists y(\varnothing\in y\;\wedge\;\forall x(x\in y\rightarrow x\cup\{x\}\in y)),$$, $$\exists y(\exists z(z\in y\wedge\forall t\neg(t\in z))\;\wedge\;\forall x(x\in y\rightarrow\exists u(u\in y\wedge\forall v(v\in u \leftrightarrow v=x\vee v\in x))).$$. The Crossword Solver finds answers to classic crosswords and cryptic crossword puzzles. Suppose that $f[z]$ is a continuous functional on a metric space $Z$ and that there is an element $z_0 \in Z$ minimizing $f[z]$. Follow Up: struct sockaddr storage initialization by network format-string. Site design / logo 2023 Stack Exchange Inc; user contributions licensed under CC BY-SA. [ 1] En funktion dremot r vldefinierad nr den ger samma resultat d ingngsvrdets representativa vrde ndras utan att dess kvantitiva vrde gr det. I see "dots" in Analysis so often that I feel it could be made formal. set of natural number w is defined as. The number of diagonals only depends on the number of edges, and so it is a well-defined function on $X/E$. Furthermore, Atanassov and Gargov introduced the notion of Interval-valued intuitionistic fuzzy sets (IVIFSs) extending the concept IFS, in which, the . Similar methods can be used to solve a Fredholm integral equation of the second kind in the spectrum, that is, when the parameter $\lambda$ of the equation is equal to one of the eigen values of the kernel. L. Colin, "Mathematics of profile inversion", D.L. Learn a new word every day. We've added a "Necessary cookies only" option to the cookie consent popup, For $m,n\in \omega, m \leq n$ imply $\exists ! Take another set $Y$, and a function $f:X\to Y$. Goncharskii, A.S. Leonov, A.G. Yagoda, "On the residual principle for solving nonlinear ill-posed problems", V.K. Definition of ill-defined: not easy to see or understand The property's borders are ill-defined. It is not well-defined because $f(1/2) = 2/2 =1$ and $f(2/4) = 3/4$. For a concrete example, the linear form $f$ on ${\mathbb R}^2$ defined by $f(1,0)=1$, $f(0,1)=-1$ and $f(-3,2)=0$ is ill-defined. Kids Definition. It was last seen in British general knowledge crossword. Secondly notice that I used "the" in the definition. The problem \ref{eq2} then is ill-posed. Enter a Crossword Clue Sort by Length It's also known as a well-organized problem. Proceedings of the 34th Midwest Instruction and Computing Symposium, University of Northern Iowa, April, 2001. This is a regularizing minimizing sequence for the functional $f_\delta[z]$ (see [TiAr]), consequently, it converges as $n \rightarrow \infty$ to an element $z_0$. The function $\phi(\alpha)$ is monotone and semi-continuous for every $\alpha > 0$. Nonlinear algorithms include the . An example of a function that is well-defined would be the function As we know, the full name of Maths is Mathematics. Select one of the following options. It is only after youve recognized the source of the problem that you can effectively solve it. As a result, students developed empirical and critical-thinking skills, while also experiencing the use of programming as a tool for investigative inquiry. Possible solutions must be compared and cross examined, keeping in mind the outcomes which will often vary depending on the methods employed. Other ill-posed problems are the solution of systems of linear algebraic equations when the system is ill-conditioned; the minimization of functionals having non-convergent minimizing sequences; various problems in linear programming and optimal control; design of optimal systems and optimization of constructions (synthesis problems for antennas and other physical systems); and various other control problems described by differential equations (in particular, differential games). In fact, Euclid proves that given two circles, this ratio is the same. A partial differential equation whose solution does not depend continuously on its parameters (including but not limited to boundary conditions) is said to be ill-posed. The main goal of the present study was to explore the role of sleep in the process of ill-defined problem solving. Vasil'ev, "The posing of certain improper problems of mathematical physics", A.N. where $\epsilon(\delta) \rightarrow 0$ as $\delta \rightarrow 0$? Check if you have access through your login credentials or your institution to get full access on this article. In mathematics, an expression is well-defined if it is unambiguous and its objects are independent of their representation. $$ But we also must make sure that the choice of $c$ is irrelevant, that is: Whenever $g(c)=g(c')$ it must also be true that $h(c)=h(c')$. Here are seven steps to a successful problem-solving process. Ill-defined. The problem of determining a solution $z=R(u)$ in a metric space $Z$ (with metric $\rho_Z(,)$) from "initial data" $u$ in a metric space $U$ (with metric $\rho_U(,)$) is said to be well-posed on the pair of spaces $(Z,U)$ if: a) for every $u \in U$ there exists a solution $z \in Z$; b) the solution is uniquely determined; and c) the problem is stable on the spaces $(Z,U)$, i.e. Tichy, W. (1998). These example sentences are selected automatically from various online news sources to reflect current usage of the word 'ill-defined.' Mathematics Stack Exchange is a question and answer site for people studying math at any level and professionals in related fields. Problems for which at least one of the conditions below, which characterize well-posed problems, is violated. So, $f(x)=\sqrt{x}$ is ''well defined'' if we specify, as an example, $f : [0,+\infty) \to \mathbb{R}$ (because in $\mathbb{R}$ the symbol $\sqrt{x}$ is, by definition the positive square root) , but, in the case $ f:\mathbb{R}\to \mathbb{C}$ it is not well defined since it can have two values for the same $x$, and becomes ''well defined'' only if we have some rule for chose one of these values ( e.g. Jordan, "Inverse methods in electromagnetics", J.R. Cann on, "The one-dimensional heat equation", Addison-Wesley (1984), A. Carasso, A.P. www.springer.com Ill-defined means that rules may or may not exist, and nobody tells you whether they do, or what they are. Problems leading to the minimization of functionals (design of antennas and other systems or constructions, problems of optimal control and many others) are also called synthesis problems. One distinguishes two types of such problems. Delivered to your inbox! Should Computer Scientists Experiment More? w = { 0, 1, 2, } = { 0, 0 +, ( 0 +) +, } (for clarity is changed to w) I agree that w is ill-defined because the " " does not specify how many steps we will go. The parameter $\alpha$ is determined from the condition $\rho_U(Az_\alpha,u_\delta) = \delta$. What is a word for the arcane equivalent of a monastery? \abs{f_\delta[z] - f[z]} \leq \delta\Omega[z]. In this definition it is not assumed that the operator $ R(u,\alpha(\delta))$ is globally single-valued. The results of previous studies indicate that various cognitive processes are . Learn how to tell if a set is well defined or not.If you want to view all of my videos in a nicely organized way, please visit https://mathandstatshelp.com/ . Third, organize your method. relationships between generators, the function is ill-defined (the opposite of well-defined). A minimizing sequence $\set{z_n}$ of $f[z]$ is called regularizing if there is a compact set $\hat{Z}$ in $Z$ containing $\set{z_n}$. Tip Four: Make the most of your Ws. Here are a few key points to consider when writing a problem statement: First, write out your vision. Az = u. It only takes a minute to sign up. More rigorously, what happens is that in this case we can ("well") define a new function $f':X/E\to Y$, as $f'([x])=f(x)$. Kryanev, "The solution of incorrectly posed problems by methods of successive approximations", M.M. Problem that is unstructured. As approximate solutions of the problems one can then take the elements $z_{\alpha_n,\delta_n}$. In this context, both the right-hand side $u$ and the operator $A$ should be among the data. If $\rho_U(u_\delta,u_T)$, then as an approximate solution of \ref{eq1} with an approximately known right-hand side $u_\delta$ one can take the element $z_\alpha = R(u_\delta,\alpha)$ obtained by means of the regularizing operator $R(u,\alpha)$, where $\alpha = \alpha(\delta)$ is compatible with the error of the initial data $u_\delta$ (see [Ti], [Ti2], [TiAr]). If the error of the right-hand side of the equation for $u_\delta$ is known, say $\rho_U(u_\delta,u_T) \leq \delta$, then in accordance with the preceding it is natural to determine $\alpha$ by the discrepancy, that is, from the relation $\rho_U(Az_\alpha^\delta,u_\delta) = \phi(\alpha) = \delta$. Empirical Investigation throughout the CS Curriculum. Is there a solutiuon to add special characters from software and how to do it, Minimising the environmental effects of my dyson brain. \label{eq2} My main area of study has been the use of . Ill-structured problems can also be considered as a way to improve students' mathematical . approximating $z_T$. When one says that something is well-defined one simply means that the definition of that something actually defines something. Compare well-defined problem. Then $R_1(u,\delta)$ is a regularizing operator for equation \ref{eq1}. Is there a proper earth ground point in this switch box? [1] il . The link was not copied. Then $R_2(u,\alpha)$ is a regularizing operator for \ref{eq1}. All Rights Reserved. &\implies h(\bar x) = h(\bar y) \text{ (In $\mathbb Z_{12}$).} They include significant social, political, economic, and scientific issues (Simon, 1973). Proceedings of the 33rd SIGCSE Technical Symposium on Computer Science Education, SIGCSE Bulletin 34(1). grammar. This is the way the set of natural numbers was introduced to me the first time I ever received a course in set theory: Axiom of Infinity (AI): There exists a set that has the empty set as one of its elements, and it is such that if $x$ is one of its elements, then $x\cup\{x\}$ is also one of its elements. The numerical parameter $\alpha$ is called the regularization parameter. College Entrance Examination Board (2001). \Omega[z] = \int_a^b (z^{\prime\prime}(x))^2 \rd x A Dictionary of Psychology , Subjects: If you preorder a special airline meal (e.g. An ill-conditioned problem is indicated by a large condition number. quotations ( mathematics) Defined in an inconsistent way. An operator $R(u,\alpha)$ from $U$ to $Z$, depending on a parameter $\alpha$, is said to be a regularizing operator (or regularization operator) for the equation $Az=u$ (in a neighbourhood of $u=u_T$) if it has the following properties: 1) there exists a $\delta_1 > 0$ such that $R(u,\alpha)$ is defined for every $\alpha$ and any $u_\delta \in U$ for which $\rho_U(u_\delta,u_T) < \delta \leq \delta_1$; and 2) there exists a function $\alpha = \alpha(\delta)$ of $\delta$ such that for any $\epsilon > 0$ there is a $\delta(\epsilon) \leq \delta_1$ such that if $u_\delta \in U$ and $\rho_U(u_\delta,u_T) \leq \delta(\epsilon)$, then $\rho_Z(z_\delta,z_T) < \epsilon$, where $z_\delta = R(u_\delta,\alpha(\delta))$. Identify those arcade games from a 1983 Brazilian music video. Tip Two: Make a statement about your issue. It is based on logical thinking, numerical calculations, and the study of shapes. One moose, two moose. This means that the statement about $f$ can be taken as a definition, what it formally means is that there exists exactly one such function (and of course it's the square root). Domains in which traditional approaches for building tutoring systems are not applicable or do not work well have been termed "ill-defined domains." This chapter provides an updated overview of the problems and solutions for building intelligent tutoring systems for these domains. Test your knowledge - and maybe learn something along the way. Today's crossword puzzle clue is a general knowledge one: Ill-defined. Overview ill-defined problem Quick Reference In the study of problem solving, any problem in which either the starting position, the allowable operations, or the goal state is not clearly specified, or a unique solution cannot be shown to exist. https://encyclopediaofmath.org/index.php?title=Ill-posed_problems&oldid=25322, Numerical analysis and scientific computing, V.Ya. The well-defined problems have specific goals, clearly . Is there a difference between non-existence and undefined? Click the answer to find similar crossword clues . Tikhonov, V.I. ensures that for the inductive set $A$, there exists a set whose elements are those elements $x$ of $A$ that have the property $P(x)$, or in other words, $\{x\in A|\;P(x)\}$ is a set. To do this, we base what we do on axioms : a mathematical argument must use the axioms clearly (with of course the caveat that people with more training are used to various things and so don't need to state the axioms they use, and don't need to go back to very basic levels when they explain their arguments - but that is a question of practice, not principle). in Share the Definition of ill on Twitter Twitter. This paper presents a methodology that combines a metacognitive model with question-prompts to guide students in defining and solving ill-defined engineering problems. To express where it is in 3 dimensions, you would need a minimum, basis, of 3 independently linear vectors, span (V1,V2,V3). Stack Exchange network consists of 181 Q&A communities including Stack Overflow, the largest, most trusted online community for developers to learn, share their knowledge, and build their careers. \newcommand{\abs}[1]{\left| #1 \right|} There is a distinction between structured, semi-structured, and unstructured problems. In the comment section of this question, Thomas Andrews say that the set $w=\{0,1,2,\cdots\}$ is ill-defined. Various physical and technological questions lead to the problems listed (see [TiAr]). Third, organize your method. In most formalisms, you will have to write $f$ in such a way that it is defined in any case; what the proof actually gives you is that $f$ is a. As IFS can represents the incomplete/ ill-defined information in a more specific manner than FST, therefore, IFS become more popular among the researchers in uncertainty modeling problems. $$ Otherwise, the expression is said to be not well defined, ill definedor ambiguous. In other words, we will say that a set $A$ is inductive if: For each $a\in A,\;a\cup\{a\}$ is also an element of $A$. To express a plane, you would use a basis (minimum number of vectors in a set required to fill the subspace) of two vectors. \end{align}. Why is the set $w={0,1,2,\ldots}$ ill-defined? Two things are equal when in every assertion each may be replaced by the other. As a result, what is an undefined problem? It appears to me that if we limit the number of $+$ to be finite, then $w=\omega_0$. In principle, they should give the precise definition, and the reason they don't is simply that they know that they could, if asked to do so, give a precise definition. If we want $w=\omega_0$ then we have to specify that there can only be finitely many $+$ above $0$. Reed, D., Miller, C., & Braught, G. (2000). You have to figure all that out for yourself. In mathematics education, problem-solving is the focus of a significant amount of research and publishing. If we want w = 0 then we have to specify that there can only be finitely many + above 0. An operator $R(u,\delta)$ from $U$ to $Z$ is said to be a regularizing operator for the equation $Az=u$ (in a neighbourhood of $u=u_T$) if it has the following properties: 1) there exists a $\delta_1 > 0$ such that the operator $R(u,\delta)$ is defined for every $\delta$, $0 \leq \delta \leq \delta_1$, and for any $u_\delta \in U$ such that $\rho_U(u_\delta,u_T) \leq \delta$; and 2) for every $\epsilon > 0$ there exists a $\delta_0 = \delta_0(\epsilon,u_T)$ such that $\rho_U(u_\delta,u_T) \leq \delta \leq \delta_0$ implies $\rho_Z(z_\delta,z_T) \leq \epsilon$, where $z_\delta = R(u_\delta,\delta)$. (Hermann Grassman Continue Reading 49 1 2 Alex Eustis Is it possible to create a concave light? The statement '' well defined'' is used in many different contexts and, generally, it means that something is defined in a way that correspond to some given ''definition'' in the specific context. $$ We have 6 possible answers in our database. Vinokurov, "On the regularization of discontinuous mappings", J. Baumeister, "Stable solution of inverse problems", Vieweg (1986), G. Backus, F. Gilbert, "The resolving power of gross earth data", J.V. There is only one possible solution set that fits this description. What courses should I sign up for? Well-defined: a problem having a clear-cut solution; can be solved by an algorithm - E.g., crossword puzzle or 3x = 2 (solve for x) Ill-defined: a problem usually having multiple possible solutions; cannot be solved by an algorithm - E.g., writing a hit song or building a career Herb Simon trained in political science; also . another set? Under these conditions, for every positive number $\delta < \rho_U(Az_0,u_\delta)$, where $z_0 \in \set{ z : \Omega[z] = \inf_{y\in F}\Omega[y] }$, there is an $\alpha(\delta)$ such that $\rho_U(Az_\alpha^\delta,u_\delta) = \delta$ (see [TiAr]). He's been ill with meningitis. In particular, the definitions we make must be "validated" from the axioms (by this I mean : if we define an object and assert its existence/uniqueness - you don't need axioms to say "a set is called a bird if it satisfies such and such things", but doing so will not give you the fact that birds exist, or that there is a unique bird). 1 Introduction Domains where classical approaches for building intelligent tutoring systems (ITS) are not applicable or do not work well have been termed "ill-defined domains" [1]. An example of something that is not well defined would for instance be an alleged function sending the same element to two different things. General Topology or Point Set Topology. Morozov, "Methods for solving incorrectly posed problems", Springer (1984) (Translated from Russian), F. Natterer, "Error bounds for Tikhonov regularization in Hilbert scales", F. Natterer, "The mathematics of computerized tomography", Wiley (1986), A. Neubauer, "An a-posteriori parameter choice for Tikhonov regularization in Hilbert scales leading to optimal convergence rates", L.E. Emerging evidence suggests that these processes also support the ability to effectively solve ill-defined problems which are those that do not have a set routine or solution. M^\alpha[z,u_\delta] = \rho_U^2(Az,u_\delta) + \alpha \Omega[z]. The existence of such an element $z_\delta$ can be proved (see [TiAr]). $$w=\{0,1,2,\cdots\}=\{0,0^+,(0^{+})^+,\cdots\}$$. Therefore, as approximate solutions of such problems one can take the values of the functional $f[z]$ on any minimizing sequence $\set{z_n}$. Suppose that $Z$ is a normed space. Lavrent'ev] Lavrentiev, "Some improperly posed problems of mathematical physics", Springer (1967) (Translated from Russian), R. Lattes, J.L. A well-defined problem, according to Oxford Reference, is a problem where the initial state or starting position, allowable operations, and goal state are all clearly specified. It deals with logical reasoning and quantitative calculation, and its development has involved an increasing degree of idealization and abstraction of its subject matter. More simply, it means that a mathematical statement is sensible and definite. $$w=\{0,1,2,\cdots\}=\{0,0^+,(0^{+})^+,\cdots\}$$ A variant of this method in Hilbert scales has been developed in [Na] with parameter choice rules given in [Ne]. Example: In the given set of data: 2, 4, 5, 5, 6, 7, the mode of the data set is 5 since it has appeared in the set twice. In simplest terms, $f:A \to B$ is well-defined if $x = y$ implies $f(x) = f(y)$. As these successes may be applicable to ill-defined domains, is important to investigate how to apply tutoring paradigms for tasks that are ill-defined. Since $\rho_U(Az_T,u_\delta) \leq \delta$, the approximate solution of $Az = u_\delta$ is looked for in the class $Z_\delta$ of elements $z_\delta$ such that $\rho_U(u_\delta,u_T) \leq \delta$. In a physical experiment the quantity $z$ is frequently inaccessible to direct measurement, but what is measured is a certain transform $Az=u$ (also called outcome). .staff with ill-defined responsibilities. The words at the top of the list are the ones most associated with ill defined, and as you go down the relatedness becomes more slight. Now in ZF ( which is the commonly accepted/used foundation for mathematics - with again, some caveats) there is no axiom that says "if OP is pretty certain of what they mean by $$, then it's ok to define a set using $$" - you can understand why.
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