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. Lavrent'ev] Lavrentiev, "Some improperly posed problems of mathematical physics", Springer (1967) (Translated from Russian), R. Lattes, J.L. How can I say the phrase "only finitely many. Answers to these basic questions were given by A.N. As a less silly example, you encounter this kind of difficulty when defining application on a tensor products by assigning values on elementary tensors and extending by linearity, since elementary tensors only span a tensor product and are far from being a basis (way too huge family). poorly stated or described; "he confuses the reader with ill-defined terms and concepts". A number of problems important in practice leads to the minimization of functionals $f[z]$. (That's also our interest on this website (complex, ill-defined, and non-immediate) CIDNI problems.) More rigorously, what happens is that in this case we can ("well") define a new function $f':X/E\to Y$, as $f'([x])=f(x)$. adjective. Science and technology However, for a non-linear operator $A$ the equation $\phi(\alpha) = \delta$ may have no solution (see [GoLeYa]). Evidently, $z_T = A^{-1}u_T$, where $A^{-1}$ is the operator inverse to $A$. $f\left(\dfrac 13 \right) = 4$ and It is based on logical thinking, numerical calculations, and the study of shapes. - Provides technical . We can then form the quotient $X/E$ (set of all equivalence classes). Ill-defined means that rules may or may not exist, and nobody tells you whether they do, or what they are. In this case $A^{-1}$ is continuous on $M$, and if instead of $u_T$ an element $u_\delta$ is known such that $\rho_U(u_\delta,u_T) \leq \delta$ and $u_\delta \in AM$, then as an approximate solution of \ref{eq1} with right-hand side $u = u_\delta$ one can take $z_\delta = A^{-1}u_\delta $. Problems with unclear goals, solution paths, or expected solutions are known as ill-defined problems. 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 $F_{1,\delta}$. Deconvolution is ill-posed and will usually not have a unique solution even in the absence of noise. How should the relativized Kleene pointclass $\Sigma^1_1(A)$ be defined? Copyright 2023 ACM, Inc. Journal of Computing Sciences in Colleges. We call $y \in \mathbb {R}$ the square root of $x$ if $y^2 = x$, and we denote it $\sqrt x$. Reed, D., Miller, C., & Braught, G. (2000). Kryanev, "The solution of incorrectly posed problems by methods of successive approximations", M.M. 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. on the quotient $G/H$ by defining $[g]*[g']=[g*g']$. By clicking Accept all cookies, you agree Stack Exchange can store cookies on your device and disclose information in accordance with our Cookie Policy. Definition of "well defined" in mathematics, We've added a "Necessary cookies only" option to the cookie consent popup. adjective. @Arthur So could you write an answer about it? Another example: $1/2$ and $2/4$ are the same fraction/equivalent. He is critically (= very badly) ill in hospital. A Computer Science Tapestry (2nd ed.). Goncharskii, A.S. Leonov, A.G. Yagoda, "On the residual principle for solving nonlinear ill-posed problems", V.K. 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 $\hat{Z}$. It appears to me that if we limit the number of $+$ to be finite, then $w=\omega_0$. ill-defined. 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). \rho_U(A\tilde{z},Az_T) \leq \delta 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). The N,M,P represent numbers from a given set. Groetsch, "The theory of Tikhonov regularization for Fredholm equations of the first kind", Pitman (1984), C.W. Enter the length or pattern for better results. In fact, ISPs frequently have unstated objectives and constraints that must be determined by the people who are solving the problem. $$ Do any two ill-founded models of set theory with order isomorphic ordinals have isomorphic copies of L? Mode Definition in Statistics A mode is defined as the value that has a higher frequency in a given set of values. 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). Proceedings of the 31st SIGCSE Technical Symposium on Computer Science Education, SIGCSE Bulletin 32(1), 202-206. Subscribe to America's largest dictionary and get thousands more definitions and advanced searchad free! Document the agreement(s). \norm{\bar{z} - z_0}_Z = \inf_{z \in Z} \norm{z - z_0}_Z . Under these conditions equation \ref{eq1} does not have a classical solution. Suppose that $Z$ is a normed space. No, leave fsolve () aside. Otherwise, a solution is called ill-defined . College Entrance Examination Board, New York, NY. An example of something that is not well defined would for instance be an alleged function sending the same element to two different things. $f\left(\dfrac xy \right) = x+y$ is not well-defined an ill-defined mission. The plant can grow at a rate of up to half a meter per year. Ivanov, "On linear problems which are not well-posed", A.V. If it is not well-posed, it needs to be re-formulated for numerical treatment. 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}$? the principal square root). If we want w = 0 then we have to specify that there can only be finitely many + above 0. &\implies 3x \equiv 3y \pmod{12}\\ An element $z_\delta$ is a solution to the problem of minimizing $\Omega[z]$ given $\rho_U(Az,u_\delta)=\delta$, that is, a solution of a problem of conditional extrema, which can be solved using Lagrange's multiplier method and minimization of the functional il . The element $z_\alpha$ minimizing $M^\alpha[z,u_\delta]$ can be regarded as the result of applying to the right-hand side of the equation $Az = u_\delta$ a certain operator $R_2(u_\delta,\alpha)$ depending on $\alpha$, that is, $z_\alpha = R_2(u_\delta,\alpha)$ in which $\alpha$ is determined by the discrepancy relation $\rho_U(Az_\alpha,u_\delta) = \delta$. The well-defined problems have specific goals, clearly . Once we have this set, and proved its properties, we can allow ourselves to write things such as $\{u_0, u_1,u_2,\}$, but that's just a matter of convenience, and in principle this should be defined precisely, referring to specific axioms/theorems. And in fact, as it was hinted at in the comments, the precise formulation of these "$$" lies in the axiom of infinity : it is with this axiom that we can make things like "$0$, then $1$, then $2$, and for all $n$, $n+1$" precise. Gestalt psychologists find it is important to think of problems as a whole. 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. Learner-Centered Assessment on College Campuses. The function $\phi(\alpha)$ is monotone and semi-continuous for every $\alpha > 0$. The result is tutoring services that exceed what was possible to offer with each individual approach for this domain. A well-defined and ill-defined problem example would be the following: If a teacher who is teaching French gives a quiz that asks students to list the 12 calendar months in chronological order in . There are also other methods for finding $\alpha(\delta)$. ArseninA.N. 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. Developing Reflective Judgment: Understanding and Promoting Intellectual Growth and Critical Thinking in Adolescents and Adults. 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. Make it clear what the issue is. Otherwise, the expression is said to be not well defined, ill defined or ambiguous. And it doesn't ensure the construction. $f\left(\dfrac 26 \right) = 8.$, The function $g:\mathbb Q \to \mathbb Z$ defined by If you preorder a special airline meal (e.g. 2002 Advanced Placement Computer Science Course Description. PRINTED FROM OXFORD REFERENCE (www.oxfordreference.com). : For every $\epsilon > 0$ there is a $\delta(\epsilon) > 0$ such that for any $u_1, u_2 \in U$ it follows from $\rho_U(u_1,u_2) \leq \delta(\epsilon)$ that $\rho_Z(z_1,z_2) < \epsilon$, where $z_1 = R(u_1)$ and $z_2 = R(u_2)$. For the interpretation of the results it is necessary to determine $z$ from $u$, that is, to solve the equation For the construction of approximate solutions to such classes both deterministic and probability approaches are possible (see [TiAr], [LaVa]). For instance, it is a mental process in psychology and a computerized process in computer science. A minimizing sequence $\set{z_n}$ of $f[z]$ is called regularizing if there is a compact set $\hat{Z}$ in $Z$ containing $\set{z_n}$. Is this the true reason why $w$ is ill-defined? Most businesses arent sufficiently rigorous when developing new products, processes, or even businesses in defining the problems theyre trying to solve and explaining why those issues are critical. Sep 16, 2017 at 19:24. Learn more about Stack Overflow the company, and our products. The concept of a well-posed problem is due to J. Hadamard (1923), who took the point of view that every mathematical problem corresponding to some physical or technological problem must be well-posed. On the basis of these arguments one has formulated the concept (or the condition) of being Tikhonov well-posed, also called conditionally well-posed (see [La]). 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 $A$. These example sentences are selected automatically from various online news sources to reflect current usage of the word 'ill-defined.' Mathematics is the science of the connection of magnitudes. In mathematics, a well-defined expression or unambiguous expression is an expression whose definition assigns it a unique interpretation or value. Ill defined Crossword Clue The Crossword Solver found 30 answers to "Ill defined", 4 letters crossword clue. Engl, H. Gfrerer, "A posteriori parameter choice for general regularization methods for solving linear ill-posed problems", C.W. As a pointer, having the axiom of infinity being its own axiom in ZF would be rather silly if this construction was well-defined. $h:\mathbb Z_8 \to \mathbb Z_{12}$ defined by $h(\bar x) = \overline{3x}$. Is there a detailed definition of the concept of a 'variable', and why do we use them as such? You missed the opportunity to title this question 'Is "well defined" well defined? The regularization method. Prior research involving cognitive processing relied heavily on instructional subjects from the areas of math, science and technology. relationships between generators, the function is ill-defined (the opposite of well-defined). This page was last edited on 25 April 2012, at 00:23. It was last seen in British general knowledge crossword. Defined in an inconsistent way. d Why Does The Reflection Principle Fail For Infinitely Many Sentences? There can be multiple ways of approaching the problem or even recognizing it. Nevertheless, integrated STEM instruction remains ill-defined with many gaps evident in the existing research of how implementation explicitly works. \end{align}. 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. Resources for learning mathematics for intelligent people? You have to figure all that out for yourself. I don't understand how that fits with the sentence following it; we could also just pick one root each for $f:\mathbb{R}\to \mathbb{C}$, couldn't we? In fact, Euclid proves that given two circles, this ratio is the same. where $\epsilon(\delta) \rightarrow 0$ as $\delta \rightarrow 0$? $$ Presentation with pain, mass, fever, anemia and leukocytosis. 1 Introduction Domains where classical approaches for building intelligent tutoring systems (ITS) are not applicable or do not work well have been termed "ill-defined domains" [1]. M^\alpha[z,u_\delta] = \rho_U^2(Az,u_\delta) + \alpha \Omega[z]. Let $\tilde{u}$ be this approximate value. Its also known as a well-organized problem. In the scene, Charlie, the 40-something bachelor uncle is asking Jake . Proof of "a set is in V iff it's pure and well-founded". Origin of ill-defined First recorded in 1865-70 Words nearby ill-defined ill-boding, ill-bred, ill-conceived, ill-conditioned, ill-considered, ill-defined, ill-disguised, ill-disposed, Ille, Ille-et-Vilaine, illegal Allyn & Bacon, Needham Heights, MA. Select one of the following options. Exempelvis om har reella ingngsvrden . Then for any $\alpha > 0$ the problem of minimizing the functional The proposed methodology is based on the concept of Weltanschauung, a term that pertains to the view through which the world is perceived, i.e., the "worldview." $$ Typically this involves including additional assumptions, such as smoothness of solution. $$ For example we know that $\dfrac 13 = \dfrac 26.$. In simplest terms, $f:A \to B$ is well-defined if $x = y$ implies $f(x) = f(y)$. As a normal solution of a corresponding degenerate system one can take a solution $z$ of minimal norm $\norm{z}$. Axiom of infinity seems to ensure such construction is possible. A naive definition of square root that is not well-defined: let $x \in \mathbb {R}$ be non-negative. Mathematics Stack Exchange is a question and answer site for people studying math at any level and professionals in related fields. Make sure no trains are approaching from either direction, The three spectroscopy laws of Kirchhoff. 1: meant to do harm or evil. Today's crossword puzzle clue is a general knowledge one: Ill-defined. - Leads diverse shop of 7 personnel ensuring effective maintenance and operations for 17 workcenters, 6 specialties. Why is the set $w={0,1,2,\ldots}$ ill-defined? Dec 2, 2016 at 18:41 1 Yes, exactly. About an argument in Famine, Affluence and Morality. Tikhonov, "On stability of inverse problems", A.N. ill-defined, unclear adjective poorly stated or described "he confuses the reader with ill-defined terms and concepts" Wiktionary (0.00 / 0 votes) Rate this definition: ill-defined adjective Poorly defined; blurry, out of focus; lacking a clear boundary. Spangdahlem Air Base, Germany. As a result, what is an undefined problem? Math. Tip Four: Make the most of your Ws.. It ensures that the result of this (ill-defined) construction is, nonetheless, a set. For example, the problem of finding a function $z(x)$ with piecewise-continuous second-order derivative on $[a,b]$ that minimizes the functional To express a plane, you would use a basis (minimum number of vectors in a set required to fill the subspace) of two vectors. PS: I know the usual definition of $\omega_0$ as the minimal infinite ordinal. What courses should I sign up for? For non-linear operators $A$ this need not be the case (see [GoLeYa]). Accessed 4 Mar. (eds.) An ill-structured problem has no clear or immediately obvious solution. Jossey-Bass, San Francisco, CA. The term well-defined (as oppsed to simply defined) is typically used when a definition seemingly depends on a choice, but in the end does not. As $\delta \rightarrow 0$, $z_\delta$ tends to $z_T$. Poorly defined; blurry, out of focus; lacking a clear boundary. Tip Two: Make a statement about your issue. ill deeds. $$ He's been ill with meningitis. The link was not copied. Under these conditions the question can only be that of finding a "solution" of the equation The main goal of the present study was to explore the role of sleep in the process of ill-defined problem solving. EDIT At the very beginning, I have pointed out that "$\ldots$" is not something we can use to define, but "$\ldots$" is used so often in Analysis that I feel I can make it a valid definition somehow. (2000). Astrachan, O. 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. &\implies \overline{3x} = \overline{3y} \text{ (In $\mathbb Z_{12}$)}\\ $g\left(\dfrac 26 \right) = \sqrt[6]{(-1)^2}=1.$, $d(\alpha\wedge\beta)=d\alpha\wedge\beta+(-1)^{|\alpha|}\alpha\wedge d\beta$. By clicking Accept all cookies, you agree Stack Exchange can store cookies on your device and disclose information in accordance with our Cookie Policy. As a result, what is an undefined problem? What is the best example of a well structured problem? An ill-conditioned problem is indicated by a large condition number. The distinction between the two is clear (now). The best answers are voted up and rise to the top, Not the answer you're looking for? The class of problems with infinitely many solutions includes degenerate systems of linear algebraic equations. Views expressed in the examples do not represent the opinion of Merriam-Webster or its editors. (2000). Connect and share knowledge within a single location that is structured and easy to search. I see "dots" in Analysis so often that I feel it could be made formal. worse wrs ; worst wrst . A problem is defined in psychology as a situation in which one is required to achieve a goal but the resolution is unclear. In most (but not all) cases, this applies to the definition of a function $f\colon A\to B$ in terms of two given functions $g\colon C\to A$ and $h\colon C\to B$: For $a\in A$ we want to define $f(a)$ by first picking an element $c\in C$ with $g(c)=a$ and then let $f(a)=h(c)$. Boerner, A.K. ill-defined problem 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." ", M.H. 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. 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. The parameter $\alpha$ is determined from the condition $\rho_U(Az_\alpha,u_\delta) = \delta$. The Crossword Solver finds answers to classic crosswords and cryptic crossword puzzles. At the basis of the approach lies the concept of a regularizing operator (see [Ti2], [TiAr]). Problems with unclear goals, solution paths, or expected solutions are known as ill-defined problems. For such problems it is irrelevant on what elements the required minimum is attained. E.g., the minimizing sequences may be divergent. Shishalskii, "Ill-posed problems of mathematical physics and analysis", Amer. 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 $\delta^2 / \beta_1(\delta) \leq \alpha \leq \beta_2(\delta)$. Payne, "Improperly posed problems in partial differential equations", SIAM (1975), B.L. In formal language, this can be translated as: $$\exists y(\varnothing\in y\;\wedge\;\forall x(x\in y\rightarrow x\cup\{x\}\in y)),$$, $$\exists y(\exists z(z\in y\wedge\forall t\neg(t\in z))\;\wedge\;\forall x(x\in y\rightarrow\exists u(u\in y\wedge\forall v(v\in u \leftrightarrow v=x\vee v\in x))).$$. www.springer.com An ill-defined problem is one in which the initial state, goal state, and/or methods are ill-defined. Sometimes, because there are had been ill for some years. If "dots" are not really something we can use to define something, then what notation should we use instead? What exactly is Kirchhoffs name? It identifies the difference between a process or products current (problem) and desired (goal) state. A quasi-solution of \ref{eq1} on $M$ is an element $\tilde{z}\in M$ that minimizes for a given $\tilde{u}$ the functional $\rho_U(Az,\tilde{u})$ on $M$ (see [Iv2]). This can be done by using stabilizing functionals $\Omega[z]$. Various physical and technological questions lead to the problems listed (see [TiAr]). Get help now: A adjective. 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. $$ Why is this sentence from The Great Gatsby grammatical? I cannot understand why it is ill-defined before we agree on what "$$" means. If we use infinite or even uncountable many $+$ then $w\neq \omega_0=\omega$. Romanov, S.P. SIGCSE Bulletin 29(4), 22-23. What is the appropriate action to take when approaching a railroad. It is the value that appears the most number of times. If we use infinite or even uncountable . The construction of regularizing operators. Tikhonov (originator), which appeared in Encyclopedia of Mathematics - ISBN 1402006098. Copyright HarperCollins Publishers 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. What is the best example of a well-structured problem, in addition? - Henry Swanson Feb 1, 2016 at 9:08 Therefore, as approximate solutions of such problems one can take the values of the functional $f[z]$ on any minimizing sequence $\set{z_n}$. Obviously, in many situation, the context is such that it is not necessary to specify all these aspect of the definition, and it is sufficient to say that the thing we are defining is '' well defined'' in such a context. 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. Proceedings of the 33rd SIGCSE Technical Symposium on Computer Science Education, SIGCSE Bulletin 34(1). Students are confronted with ill-structured problems on a regular basis in their daily lives. In this case, Monsieur Poirot can't reasonably restrict the number of suspects before he does a bit of legwork. Is there a proper earth ground point in this switch box? Mathematicians often do this, however : they define a set with $$ or a sequence by giving the first few terms and saying that "the pattern is obvious" : again, this is a matter of practice, not principle. [Gr]); for choices of the regularization parameter leading to optimal convergence rates for such methods see [EnGf]. an ill-defined mission Dictionary Entries Near ill-defined ill-deedie ill-defined ill-disposed See More Nearby Entries Cite this Entry Style "Ill-defined." Delivered to your inbox! We use cookies to ensure that we give you the best experience on our website. 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. If we want $w=\omega_0$ then we have to specify that there can only be finitely many $+$ above $0$. At heart, I am a research statistician. Vasil'ev, "The posing of certain improper problems of mathematical physics", A.N. &\implies x \equiv y \pmod 8\\ 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. 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. Is the term "properly defined" equivalent to "well-defined"? Why would this make AoI pointless? Evaluate the options and list the possible solutions (options). Walker, H. (1997). In these problems one cannot take as approximate solutions the elements of minimizing sequences. Linear deconvolution algorithms include inverse filtering and Wiener filtering. I had the same question years ago, as the term seems to be used a lot without explanation. The, Pyrex glass is dishwasher safe, refrigerator safe, microwave safe, pre-heated oven safe, and freezer safe; the lids are BPA-free, dishwasher safe, and top-rack dishwasher and, Slow down and be prepared to come to a halt when approaching an unmarked railroad crossing. But if a set $x$ has the property $P(x)$, then we have that it is an element of every inductive set, and, in particular, is an element of the inductive set $A$, so every natural number belongs to $A$ and: $$\{x\in A|\; P(x)\}=\{x| x\text{ is an element of every inductive set}\}=\{x| x\text{ is a natural number}\}$$, $\qquad\qquad\qquad\qquad\qquad\qquad\qquad\qquad\qquad\qquad\qquad\qquad\qquad\qquad\qquad\qquad\qquad\qquad\qquad\square$. In your case, when we're very clearly at the beginning of learning formal mathematics, it is not clear that you could give a precise formulation of what's hidden in those "$$". 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. Huba, M.E., & Freed, J.E. In mathematics, an expression is well-defined if it is unambiguous and its objects are independent of their representation. This put the expediency of studying ill-posed problems in doubt. Braught, G., & Reed, D. (2002). Frequently, instead of $f[z]$ one takes its $\delta$-approximation $f_\delta[z]$ relative to $\Omega[z]$, that is, a functional such that for every $z \in F_1$, Click the answer to find similar crossword clues . 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.