For example, the problem of finding a function $z(x)$ with piecewise-continuous second-order derivative on $[a,b]$ that minimizes the functional Thus, the task of finding approximate solutions of \ref{eq1} that are stable under small changes of the right-hand side reduces to: a) finding a regularizing operator; and b) determining the regularization parameter $\alpha$ from additional information on the problem, for example, the size of the error with which the right-hand side $u$ is given. $$ June 29, 2022 Posted in&nbspkawasaki monster energy jersey. Stack Exchange network consists of 181 Q&A communities including Stack Overflow, the largest, most trusted online community for developers to learn, share their knowledge, and build their careers. What is the best example of a well structured problem? Beck, B. Blackwell, C.R. Here are seven steps to a successful problem-solving process. A typical example is the problem of overpopulation, which satisfies none of these criteria. Prior research involving cognitive processing relied heavily on instructional subjects from the areas of math, science and technology. If there is an $\alpha$ for which $\rho_U(Az_\alpha,u_\delta) = \delta$, then the original variational problem is equivalent to that of minimizing $M^\alpha[z,u_\delta]$, which can be solved by various methods on a computer (for example, by solving the corresponding Euler equation for $M^\alpha[z,u_\delta]$). I must be missing something; what's the rule for choosing $f(25) = 5$ or $f(25) = -5$ if we define $f: [0, +\infty) \to \mathbb{R}$? Vasil'ev, "The posing of certain improper problems of mathematical physics", A.N. $$ Lavrent'ev, V.G. The existence of such an element $z_\delta$ can be proved (see [TiAr]). An ill-defined problem is one that lacks one or more of the specified properties, and most problems encountered in everyday life fall into this category. In mathematics, an expression is well-defined if it is unambiguous and its objects are independent of their representation. What do you mean by ill-defined? Key facts. Select one of the following options. Inom matematiken innebr vldefinierad att definitionen av ett uttryck har en unik tolkning eller ger endast ett vrde. worse wrs ; worst wrst . because $$ Similarly approximate solutions of ill-posed problems in optimal control can be constructed. It can be regarded as the result of applying a certain operator $R_1(u_\delta,d)$ to the right-hand side of the equation $Az = u_\delta$, that is, $z_\delta=R_1(u_\delta,d)$. By clicking Accept all cookies, you agree Stack Exchange can store cookies on your device and disclose information in accordance with our Cookie Policy. ($F_1$ can be the whole of $Z$.) $f\left(\dfrac 13 \right) = 4$ and The existence of quasi-solutions is guaranteed only when the set $M$ of possible solutions is compact. Two problems arise with this: First of all, we must make sure that for each $a\in A$ there exists $c\in C$ with $g(c)=a$, in other words: $g$ must be surjective. Ill-defined means that rules may or may not exist, and nobody tells you whether they do, or what they are. Stone, "Improperly posed boundary value problems", Pitman (1975), A.M. Cormak, "Representation of a function by its line integrals with some radiological applications". Computer science has really changed the conceptual difficulties in acquiring mathematics knowledge. Ill-defined definition: If you describe something as ill-defined , you mean that its exact nature or extent is. 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). Semi structured problems are defined as problems that are less routine in life. Kryanev, "The solution of incorrectly posed problems by methods of successive approximations", M.M. Various physical and technological questions lead to the problems listed (see [TiAr]). An ill-conditioned problem is indicated by a large condition number. We focus on the domain of intercultural competence, where . in Typically this involves including additional assumptions, such as smoothness of solution. Tichy, W. (1998). See also Ambiguous, Ill-Posed , Well-Defined Explore with Wolfram|Alpha More things to try: partial differential equations 4x+3=19 conjugate: 1+3i+4j+3k, 1+-1i-j+3k Cite this as: Weisstein, Eric W. "Ill-Defined." Bakushinskii, "A general method for constructing regularizing algorithms for a linear ill-posed equation in Hilbert space", A.V. The parameter choice rule discussed in the article given by $\rho_U(Az_\alpha^\delta,u_\delta) = \delta$ is called the discrepancy principle ([Mo]), or often the Morozov discrepancy principle. rev2023.3.3.43278. Is there a solutiuon to add special characters from software and how to do it, Minimising the environmental effects of my dyson brain. 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 . Secondly notice that I used "the" in the definition. Linear deconvolution algorithms include inverse filtering and Wiener filtering. Many problems in the design of optimal systems or constructions fall in this class. It is assumed that the equation $Az = u_T$ has a unique solution $z_T$. What is the best example of a well structured problem? c: not being in good health. 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. Let $\set{\delta_n}$ and $\set{\alpha_n}$ be null-sequences such that $\delta_n/\alpha_n \leq q < 1$ for every $n$, and let $\set{z_{\alpha_n,\delta_n}} $ be a sequence of elements minimizing $M^{\alpha_n}[z,f_{\delta_n}]$. Problem-solving is the subject of a major portion of research and publishing in mathematics education. The PISA and TIMSS show that Korean students have difficulty solving problems that connect mathematical concepts with everyday life. The exterior derivative on $M$ is a $\mathbb{R}$ linear map $d:\Omega^*(M)\to\Omega^{*+1}(M)$ such that. Under these conditions the question can only be that of finding a "solution" of the equation Discuss contingencies, monitoring, and evaluation with each other. I have a Psychology Ph.D. focusing on Mathematical Psychology/Neuroscience and a Masters in Statistics. The ill-defined problemsare those that do not have clear goals, solution paths, or expected solution. Therefore, as approximate solutions of such problems one can take the values of the functional $f[z]$ on any minimizing sequence $\set{z_n}$. Why Does The Reflection Principle Fail For Infinitely Many Sentences? To save this word, you'll need to log in. If you preorder a special airline meal (e.g. Why does Mister Mxyzptlk need to have a weakness in the comics? The selection method. What can a lawyer do if the client wants him to be acquitted of everything despite serious evidence? A operator is well defined if all N,M,P are inside the given set. A typical mathematical (2 2 = 4) question is an example of a well-structured problem. PRINTED FROM OXFORD REFERENCE (www.oxfordreference.com). Suppose that $Z$ is a normed space. Walker, H. (1997). $$w=\{0,1,2,\cdots\}=\{0,0^+,(0^{+})^+,\cdots\}$$ For instance, it is a mental process in psychology and a computerized process in computer science. To express where it is in 3 dimensions, you would need a minimum, basis, of 3 independently linear vectors, span (V1,V2,V3). given the function $f(x)=\sqrt{x}=y$ such that $y^2=x$. Then for any $\alpha > 0$ the problem of minimizing the functional College Entrance Examination Board (2001). Mathematics is the science of the connection of magnitudes. The result is tutoring services that exceed what was possible to offer with each individual approach for this domain. From: Is there a difference between non-existence and undefined? what is something? A Dictionary of Psychology , Subjects: Don't be surprised if none of them want the spotl One goose, two geese. NCAA News, March 12, 2001. http://www.ncaa.org/news/2001/20010312/active/3806n11.html. Aug 2008 - Jul 20091 year. $g\left(\dfrac 13 \right) = \sqrt[3]{(-1)^1}=-1$ and It deals with logical reasoning and quantitative calculation, and its development has involved an increasing degree of idealization and abstraction of its subject matter. Answers to these basic questions were given by A.N. 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. About an argument in Famine, Affluence and Morality. Women's volleyball committees act on championship issues. See also Ill-Defined, Well-Defined Explore with Wolfram|Alpha More things to try: Beta (5, 4) feigenbaum alpha Cite this as: Structured problems are simple problems that can be determined and solved by repeated examination and testing of the problems. In particular, a function is well-defined if it gives the same result when the form but not the value of an input is changed. Make it clear what the issue is. If $f(x)=f(y)$ whenever $x$ and $y$ belong to the same equivalence class, then we say that $f$ is well-defined on $X/E$, which intuitively means that it depends only on the class. Boerner, A.K. The distinction between the two is clear (now). The problem \ref{eq2} then is ill-posed. For non-linear operators $A$ this need not be the case (see [GoLeYa]). Jossey-Bass, San Francisco, CA. and takes given values $\set{z_i}$ on a grid $\set{x_i}$, is equivalent to the construction of a spline of the second degree. Why are physically impossible and logically impossible concepts considered separate in terms of probability? Az = u. We call $y \in \mathbb {R}$ the square root of $x$ if $y^2 = x$, and we denote it $\sqrt x$. Here are the possible solutions for "Ill-defined" clue. In mathematics, a well-defined expressionor unambiguous expressionis an expressionwhose definition assigns it a unique interpretation or value. To manage your alert preferences, click on the button below. The Tower of Hanoi, the Wason selection task, and water-jar issues are all typical examples. Can archive.org's Wayback Machine ignore some query terms? Such problems are called essentially ill-posed. 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. A problem statement is a short description of an issue or a condition that needs to be addressed. The fascinating story behind many people's favori Can you handle the (barometric) pressure? The definition itself does not become a "better" definition by saying that $f$ is well-defined. An ill-structured problem has no clear or immediately obvious solution. 2001-2002 NAGWS Official Rules, Interpretations & Officiating Rulebook. Whenever a mathematical object is constructed there is need for convincing arguments that the construction isn't ambigouos. The following are some of the subfields of topology. ', which I'm sure would've attracted many more votes via Hot Network Questions. The real reason it is ill-defined is that it is ill-defined ! As approximate solutions of the problems one can then take the elements $z_{\alpha_n,\delta_n}$. \newcommand{\set}[1]{\left\{ #1 \right\}} This paper describes a specific ill-defined problem that was successfully used as an assignment in a recent CS1 course. Most common presentation: ill-defined osteolytic lesion with multiple small holes in the diaphysis of a long bone in a child with a large soft tissue mass. Since $u_T$ is obtained by measurement, it is known only approximately. But how do we know that this does not depend on our choice of circle? Intelligent tutoring systems have increased student learning in many domains with well-structured tasks such as math and science. In fact: a) such a solution need not exist on $Z$, since $\tilde{u}$ need not belong to $AZ$; and b) such a solution, if it exists, need not be stable under small changes of $\tilde{u}$ (due to the fact that $A^{-1}$ is not continuous) and, consequently, need not have a physical interpretation. ill. 1 of 3 adjective. Sep 16, 2017 at 19:24. \rho_U(u_\delta,u_T) \leq \delta, \qquad Third, organize your method. Is there a detailed definition of the concept of a 'variable', and why do we use them as such? Discuss contingencies, monitoring, and evaluation with each other. Symptoms, Signs, and Ill-Defined Conditions (780-799) This section contains symptoms, signs, abnormal laboratory or other investigative procedures results, and ill-defined conditions for which no diagnosis is recorded elsewhere. | Meaning, pronunciation, translations and examples Definition of "well defined" in mathematics, We've added a "Necessary cookies only" option to the cookie consent popup. $$. Resources for learning mathematics for intelligent people? Do any two ill-founded models of set theory with order isomorphic ordinals have isomorphic copies of L? adjective. A solution to a partial differential equation that is a continuous function of its values on the boundary is said to be well-defined. \rho_U^2(A_hz,u_\delta) = \bigl( \delta + h \Omega[z_\alpha]^{1/2} \bigr)^2. For any positive number $\epsilon$ and functions $\beta_1(\delta)$ and $\beta_2(\delta)$ from $T_{\delta_1}$ such that $\beta_2(0) = 0$ and $\delta^2 / \beta_1(\delta) \leq \beta_2(\delta)$, there exists a $\delta_0 = \delta_0(\epsilon,\beta_1,\beta_2)$ such that for $u_\delta \in U$ and $\delta \leq \delta_0$ it follows from $\rho_U(u_\delta,u_T) \leq \delta$ that $\rho_Z(z^\delta,z_T) \leq \epsilon$, where $z^\alpha = R_2(u_\delta,\alpha)$ for all $\alpha$ for which $\delta^2 / \beta_1(\delta) \leq \alpha \leq \beta_2(\delta)$. Are there tables of wastage rates for different fruit and veg? Definition. ill-defined. My code is GPL licensed, can I issue a license to have my code be distributed in a specific MIT licensed project? Lavrent'ev] Lavrentiev, "Some improperly posed problems of mathematical physics", Springer (1967) (Translated from Russian), R. Lattes, J.L. For the interpretation of the results it is necessary to determine $z$ from $u$, that is, to solve the equation PS: I know the usual definition of $\omega_0$ as the minimal infinite ordinal. The best answers are voted up and rise to the top, Not the answer you're looking for? 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. Ill-defined. How to show that an expression of a finite type must be one of the finitely many possible values? Connect and share knowledge within a single location that is structured and easy to search. ill-defined adjective : not easy to see or understand The property's borders are ill-defined. Should Computer Scientists Experiment More? Background:Ill-structured problems are contextualized, require learners to define the problems as well as determine the information and skills needed to solve them. Winning! A broad class of so-called inverse problems that arise in physics, technology and other branches of science, in particular, problems of data processing of physical experiments, belongs to the class of ill-posed problems. However, I don't know how to say this in a rigorous way. Problems that are well-defined lead to breakthrough solutions. 'Hiemal,' 'brumation,' & other rare wintry words. Is it possible to create a concave light? In many cases the operator $A$ is such that its inverse $A^{-1}$ is not continuous, for example, when $A$ is a completely-continuous operator in a Hilbert space, in particular an integral operator of the form Understand everyones needs. $$0=\emptyset,\ 1=0^+,\ 2=1^+,\ \cdots$$ If the conditions don't hold, $f$ is not somehow "less well defined", it is not defined at all. Get help now: A satisfies three properties above. adjective If you describe something as ill-defined, you mean that its exact nature or extent is not as clear as it should be or could be. [Gr]); for choices of the regularization parameter leading to optimal convergence rates for such methods see [EnGf]. Disequilibration for Teaching the Scientific Method in Computer Science. $f\left(\dfrac 26 \right) = 8.$, The function $g:\mathbb Q \to \mathbb Z$ defined by This is a regularizing minimizing sequence for the functional $f_\delta[z]$ (see [TiAr]), consequently, it converges as $n \rightarrow \infty$ to an element $z_0$. If it is not well-posed, it needs to be re-formulated for numerical treatment. You might explain that the reason this comes up is that often classes (i.e. An approximation to a normal solution that is stable under small changes in the right-hand side of \ref{eq1} can be found by the regularization method described above. We have 6 possible answers in our database. In the smoothing functional one can take for $\Omega[z]$ the functional $\Omega[z] = \norm{z}^2$. Understand everyones needs. By clicking Accept all cookies, you agree Stack Exchange can store cookies on your device and disclose information in accordance with our Cookie Policy. In this context, both the right-hand side $u$ and the operator $A$ should be among the data. This is important. 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. Let $f(x)$ be a function defined on $\mathbb R^+$ such that $f(x)>0$ and $(f(x))^2=x$, then $f$ is well defined. $g\left(\dfrac mn \right) = \sqrt[n]{(-1)^m}$ So-called badly-conditioned systems of linear algebraic equations can be regarded as systems obtained from degenerate ones when the operator $A$ is replaced by its approximation $A_h$. As an approximate solution one takes then a generalized solution, a so-called quasi-solution (see [Iv]). In mathematics (and in this case in particular), an operation (which is a type of function), such as $+,-,\setminus$ is a relation between two sets (domain/codomain), so it does not change the domain in any way. The formal mathematics problem makes the excuse that mathematics is dry, difficult, and unattractive, and some students assume that mathematics is not related to human activity. Is it suspicious or odd to stand by the gate of a GA airport watching the planes? $$ This set is unique, by the Axiom of Extensionality, and is the set of the natural numbers, which we represent by $\mathbb{N}$. He is critically (= very badly) ill in hospital. Document the agreement(s). Mathematics Stack Exchange is a question and answer site for people studying math at any level and professionals in related fields. $$ The class of problems with infinitely many solutions includes degenerate systems of linear algebraic equations. Why is this sentence from The Great Gatsby grammatical? If $A$ is a linear operator, $Z$ a Hilbert space and $\Omega[z]$ a strictly-convex functional (for example, quadratic), then the element $z_{\alpha_\delta}$ is unique and $\phi(\alpha)$ is a single-valued function. In these problems one cannot take as approximate solutions the elements of minimizing sequences. It ensures that the result of this (ill-defined) construction is, nonetheless, a set. Document the agreement(s). Otherwise, the expression is said to be not well defined, ill definedor ambiguous. Evaluate the options and list the possible solutions (options). If $A$ is a bounded linear operator between Hilbert spaces, then, as also mentioned above, regularization operators can be constructed viaspectral theory: If $U(\alpha,\lambda) \rightarrow 1/\lambda$ as $\alpha \rightarrow 0$, then under mild assumptions, $U(\alpha,A^*A)A^*$ is a regularization operator (cf. Payne, "Improperly posed problems in partial differential equations", SIAM (1975), B.L. It's also known as a well-organized problem. To repeat: After this, $f$ is in fact defined. 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. Furthermore, competing factors may suggest several approaches to the problem, requiring careful analysis to determine the best approach. For example, a set that is identified as "the set of even whole numbers between 1 and 11" is a well-defined set because it is possible to identify the exact members of the set: 2, 4, 6, 8 and 10. ill-defined ( comparative more ill-defined, superlative most ill-defined ) Poorly defined; blurry, out of focus; lacking a clear boundary . There is an additional, very useful notion of well-definedness, that was not written (so far) in the other answers, and it is the notion of well-definedness in an equivalence class/quotient space. Learn a new word every day. &\implies 3x \equiv 3y \pmod{12}\\ There can be multiple ways of approaching the problem or even recognizing it. 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 $\rho_U(Az_\alpha^\delta,u_\delta) = \phi(\alpha) = \delta$. A naive definition of square root that is not well-defined: let $x \in \mathbb {R}$ be non-negative. Well-defined is a broader concept but it's when doing computations with equivalence classes via a member of them that the issue is forced and people make mistakes. What is a word for the arcane equivalent of a monastery? Enter the length or pattern for better results. For the construction of approximate solutions to such classes both deterministic and probability approaches are possible (see [TiAr], [LaVa]). Poirot is solving an ill-defined problemone in which the initial conditions and/or the final conditions are unclear. We will try to find the right answer to this particular crossword clue. M^\alpha[z,u_\delta] = \rho_U^2(Az,u_\delta) + \alpha \Omega[z]. A broad class of so-called inverse problems that arise in physics, technology and other branches of science, in particular, problems of data processing of physical experiments, belongs to the class of ill-posed problems. Sophia fell ill/ was taken ill (= became ill) while on holiday. Developing Empirical Skills in an Introductory Computer Science Course. In the comment section of this question, Thomas Andrews say that the set $w=\{0,1,2,\cdots\}$ is ill-defined. The problem statement should be designed to address the Five Ws by focusing on the facts. In fact, ISPs frequently have unstated objectives and constraints that must be determined by the people who are solving the problem. Such problems are called unstable or ill-posed. $$ Clancy, M., & Linn, M. (1992). Furthermore, Atanassov and Gargov introduced the notion of Interval-valued intuitionistic fuzzy sets (IVIFSs) extending the concept IFS, in which, the . Learn more about Stack Overflow the company, and our products. In this definition it is not assumed that the operator $ R(u,\alpha(\delta))$ is globally single-valued. There are two different types of problems: ill-defined and well-defined; different approaches are used for each. I see "dots" in Analysis so often that I feel it could be made formal. $h:\mathbb Z_8 \to \mathbb Z_{12}$ defined by $h(\bar x) = \overline{3x}$. Only if $g,h$ fulfil these conditions the above construction will actually define a function $f\colon A\to B$. Personalised Then one might wonder, Can you ship helium balloons in a box? Helium Balloons: How to Blow It Up Using an inflated Mylar balloon, Duranta erecta is a large shrub or small tree. This is ill-defined when $H$ is not a normal subgroup since the result may depend on the choice of $g$ and $g'$. Learner-Centered Assessment on College Campuses. Now, I will pose the following questions: Was it necessary at all to use any dots, at any point, in the construction of the natural numbers? had been ill for some years. [3] One of the main goals of Hilbert's program was a finitistic proof of the consistency of the axioms of arithmetic: that is his second problem. Tikhonov (originator), which appeared in Encyclopedia of Mathematics - ISBN 1402006098. The link was not copied. The numerical parameter $\alpha$ is called the regularization parameter. The regularization method is closely connected with the construction of splines (cf. 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. If the problem is well-posed, then it stands a good chance of solution on a computer using a stable algorithm. 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}. Is there a proper earth ground point in this switch box? \begin{equation} Clearly, it should be so defined that it is stable under small changes of the original information. $$w=\{0,1,2,\cdots\}=\{0,0^+,(0^{+})^+,\cdots\}$$. Test your knowledge - and maybe learn something along the way. Computer 31(5), 32-40. What exactly is Kirchhoffs name? Meaning of ill in English ill adjective uk / l / us / l / ill adjective (NOT WELL) A2 [ not usually before noun ] not feeling well, or suffering from a disease: I felt ill so I went home. $$ &\implies h(\bar x) = h(\bar y) \text{ (In $\mathbb Z_{12}$).} It is critical to understand the vision in order to decide what needs to be done when solving the problem. Consider the "function" $f: a/b \mapsto (a+1)/b$. Why are Suriname, Belize, and Guinea-Bissau classified as "Small Island Developing States"? Dari segi perumusan, cara menjawab dan kemungkinan jawabannya, masalah dapat dibedakan menjadi masalah yang dibatasi dengan baik (well-defined), dan masalah yang dibatasi tidak dengan baik. In completing this assignment, students actively participated in the entire process of problem solving and scientific inquiry, from the formulation of a hypothesis, to the design and implementation of experiments (via a program), to the collection and analysis of the experimental data. 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. Suppose that $f[z]$ is a continuous functional on a metric space $Z$ and that there is an element $z_0 \in Z$ minimizing $f[z]$. This article was adapted from an original article by V.Ya. Under these conditions the procedure for obtaining an approximate solution is the same, only instead of $M^\alpha[z,u_\delta]$ one has to consider the functional 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 $z_\delta = R(u_\delta,\delta)$. So the span of the plane would be span (V1,V2). In simplest terms, $f:A \to B$ is well-defined if $x = y$ implies $f(x) = f(y)$. this function is not well defined. Some simple and well-defined problems are known as well-structured problems, and they have a set number of possible solutions; solutions are either 100% correct or completely incorrect. A naive definition of square root that is not well-defined: let $x \in \mathbb{R}$ be non-negative. Then $R_2(u,\alpha)$ is a regularizing operator for \ref{eq1}. Engl, H. Gfrerer, "A posteriori parameter choice for general regularization methods for solving linear ill-posed problems", C.W. It is not well-defined because $f(1/2) = 2/2 =1$ and $f(2/4) = 3/4$.