Recent zbMATH articles in MSC 03Ehttps://www.zbmath.org/atom/cc/03E2022-05-16T20:40:13.078697ZWerkzeugIndestructibility of ideals and MAD familieshttps://www.zbmath.org/1483.030282022-05-16T20:40:13.078697Z"Chodounský, David"https://www.zbmath.org/authors/?q=ai:chodounsky.david"Guzmán, Osvaldo"https://www.zbmath.org/authors/?q=ai:guzman-gonzalez.osvaldoThis very nice article surveys the recent body of work devoted to forcing indestructibility of ideals and maximal almost disjoint families. Recall that a tall ideal is \emph{indestructible} by a forcing notion \(\mathbb P\) if it remains tall in any generic extension by \(\mathbb P\). Analogously, a MAD family is \emph{indestructible} by \(\mathbb P\) if it remains maximal or, equivalently, if the ideal generated by the family is indestructible by \(\mathbb P\).
The article is clearly written, contains proofs of several of the most important results, a list of the outstanding open questions, and an extensive bibliography. We recommend it to anyone interested in forcing aspects of set-theory of the reals.
Reviewer: Michael Hrusak (Morelia)The satisfiability problem for Boolean set theory with a choice correspondencehttps://www.zbmath.org/1483.030292022-05-16T20:40:13.078697Z"Cantone, Domenico"https://www.zbmath.org/authors/?q=ai:cantone.domenico"Giarlotta, Alfio"https://www.zbmath.org/authors/?q=ai:giarlotta.alfio"Watson, Stephen"https://www.zbmath.org/authors/?q=ai:watson.stephen-wSummary: Given a set \(U\) of alternatives, a choice (correspondence) on \(U\) is a contractive map \(c\) defined on a family \(\Omega\) of nonempty subsets of \(U\). Semantically, a choice \(c\) associates to each menu \(A\in\Omega\) a nonempty subset \(c(A)\subseteq A\) comprising all elements of \(A\) that are deemed selectable by an agent. A choice on \(U\) is total if its domain is the powerset of \(U\) minus the empty set, and partial otherwise. According to the theory of revealed preferences, a choice is rationalizable if it can be retrieved from a binary relation on \(U\) by taking all maximal elements of each menu. It is well-known that rationalizable choices are characterized by the satisfaction of suitable axioms of consistency, which codify logical rules of selection within menus. For instance, \textsf{WARP} (Weak Axiom of Revealed Preference) characterizes choices rationalizable by a transitive relation. Here we study the satisfiability problem for unquantified formulae of an elementary fragment of set theory involving a choice function symbol \(\mathsf{c}\), the Boolean set operators and the singleton, the equality and inclusion predicates, and the propositional connectives. In particular, we consider the cases in which the interpretation of \(\mathsf{c}\) satisfies any combination of two specific axioms of consistency, whose conjunction is equivalent to \textsf{WARP}. In two cases we prove that the related satisfiability problem is NP-complete, whereas in the remaining cases we obtain NP-completeness under the additional assumption that the number of choice terms is constant.
For the entire collection see [Zbl 1436.68017].On densely complete metric spaces and extensions of uniformly continuous functions in ZFhttps://www.zbmath.org/1483.030302022-05-16T20:40:13.078697Z"Keremedis, Kyriakos"https://www.zbmath.org/authors/?q=ai:keremedis.kyriakos"Wajch, Eliza"https://www.zbmath.org/authors/?q=ai:wajch.elizaA metric space \(X\) is called densely complete if there exists a dense set \(D\) in \(X\) such that every Cauchy sequence of points of \(D\) converges in \(X\).
In this work, the authors prove that the countable axiom of choice, CAC for abbreviation, is equivalent to the following statements:
(i) Every densely complete (connected) metric space \(X\) is complete.\
(ii) For every pair of metric spaces \(X\) and \(Y\), if \(Y\) is complete and \(S\) is a dense subspace of \(X\), while \(f : S \rightarrow Y\) is a uniformly continuous function, then there exists a uniformly continuous extension \(F : X\rightarrow Y\) of \(f\).\
(iii) Complete subspaces of metric spaces have complete closures.\
(iv) Complete subspaces of metric spaces are closed.
Also they prove that, for every positive integer \(n\), the space \(\mathbb R^n\) is sequential if and only if \(\mathbb R\) is sequential. Finally, it is shown that \(\mathbb R \times \mathbb Q\) is not densely complete if and only if \(\mathrm{CAC}(\mathbb R)\) holds.
Reviewer: Cenap Özel (İzmir)Automatic continuity of abstract homomorphisms between locally compact and Polish groupshttps://www.zbmath.org/1483.220012022-05-16T20:40:13.078697Z"Braun, O."https://www.zbmath.org/authors/?q=ai:braun.oliver|braun.oleg-m"Hofmann, Karl H."https://www.zbmath.org/authors/?q=ai:hofmann.karl-heinrich"Kramer, L."https://www.zbmath.org/authors/?q=ai:kramer.linusThe authors introduce the following class \(\mathcal{K}\) of \textit{almost Polish spaces}:
\noindent Every \(X\in\mathcal{K}\) is a Hausdorff topological space, \(\mathcal{K}\) is closed under the passage to closed subspaces and closed under finite products, and the following properties are satisfied: (1) Every open covering of \(X\) has a countable subcovering. (2) The space \(X\) is not a countable union of nowhere dense subsets. (3) For each continuous image \(A\subseteq X\) of some \(\mathcal{K}\)-member there is an open set \(U\subseteq X\) such that the symmetric difference \((A\setminus U)\cap (U\setminus A)\) is a countable union of nowhere dense subsets of \(X\).
The class \(\mathcal{P}\) of Polish spaces, the class \(\mathcal{L}^\sigma\) of locally compact \(\sigma\)-spaces and the class \(\mathcal{C}\) of compact spaces are almost Polish.
If \(\mathcal{K}\) is an almost Polish class and \(X\in\mathcal{K}\), then \(A\subseteq X\) is called a \textit{\(\mathcal{K}\)-analytic set} if \(A=\psi(Z)\) holds for some \(Z\in\mathcal{K}\) and some continuous map \(\psi:Z\to X\).
If \(G\) is a topological group with \(G\in\mathcal{K}\), then \(G\) is called \textit{\(\mathcal{K}\)-rigid} if the following holds:
\noindent For every short exact sequence of groups \[1\to N\hookrightarrow K\overset{\varphi} \to G\to 1,\] where \(\varphi\) is an abstract group homomorphism, where \(G,K\in\mathcal{K}\), and where the kernel \(N\) of \(\varphi\) is \(\mathcal{K}\)-analytic, the homomorphism \(\varphi\) is automatically continuous and open.
The authors show that rigidity fails in the following cases:
\noindent (a) Abelian groups. (b) Groups which are not locally compact or \(\sigma\)-compact. (c) Infinite products of compact Lie groups if the kernel is not restricted. (d) \(\mathrm{SL}(n,\mathbb{C})\) and all infinite complex linear algebraic groups. (e) Certain connected perfect real algebraic groups.
The first main result is Theorem 2.9. Let \(\mathcal{K}\) be a class of almost Polish spaces, and let \(K,G\in\mathcal{K}\) be topological groups. If \(\varphi:K\to G\) is an abstract group homomorphism such that for every identity neighborhood \(U\subseteq G\) there exists an identity neighborhood \(V\subseteq U\) such that \(\varphi^{-1}(V)\) is almost open (which is the case if \(\varphi^{-1}(V)\) is \(\mathcal{K}\)-analytic), then \(\varphi\) is continuous. If \(\varphi\) is in addition surjective, then it is open.
\noindent In this situation, let \(G\) be a Lie group whose Lie algebra is perfect. If \((K,\ker\varphi)\in\mathcal{K}_a\) and if there exists a compact spacious subset \(C\subseteq G\) such that \(\varphi^{-1}(C)\) is \(\mathcal{K}\)-analytic, then \(\varphi\) is continuous and open. (Let \(\mathcal{K}_a\) denote the class of all pairs \((X,A)\), where \(X\in\mathcal{K}\) and \(A\subseteq X\) is \(\mathcal{K}\)-analytic. \(C\) is called \textit{spacious} if some product of finitely many translates of \(CC^{-1}\) has nonempty interior.)
\noindent By setting \(C=G\) this result implies: Let \(G\) be a compact Lie group whose Lie algebra is semisimple. Then \(G\) is rigid within every almost Polish class \(\mathcal{K}\) that contains \(G\).
\noindent This assertion contains results of \textit{R.~R. Kallman} [Adv. Math. 12, 416--417 (1974; Zbl. 0273.22009)] and \textit{P. Gartside} and \textit{B. Pejić} [Topology Appl. 155, 992--999 (2008; Zbl. 1151.54029)].
The second main result is Theorem 4.6: Let \(G\) be a Lie group such that the center of the connected component is finite and the Lie algebra of \(G\) is a direct sum of absolutely simple ideals. If \(\mathcal{K}\) is an almost Polish class containing \(G\), then \(G\) is rigid within \(\mathcal{K}\).
Then the authors consider semiproducts of Lie groups. In Theorem 5.6 a list of such groups is given which are rigid in every Polish class \(\mathcal{K}\) containing them.
In Section 6 the rigidity of topologically finitely generated profinite groups is studied. The main result is Theorem 6.3: Let \(G\) be a topologically finitely generated profinite group, and let \(\mathcal{K}\) be an almost Polish class. If \(G\) is contained in \(\mathcal{K}\), then \(G\) is rigid within \(\mathcal{K}\).
\noindent This Theorem generalizes a result of Gartside and Pejić [loc. cit.].
Theorem 6.3 remains valid if \(G\) is a compact quasi-semisimple group (see Theorem 7.7). (A nontrivial compact group \(S\) with center \(C(S)\) is called \textit{quasi-semisimple} if the commutator subgroup is dense in \(S\) and if \(S/C(S)\) is topologically simple.)
The very interesting paper ends with Theorem 8.1: Let \(G\) be a Lie group whose Lie algebra is perfect and let \(H\) be a topological group. Let \(\psi:G\to H\) be an abstract homomorphism such that there exists a compact spacious set \(C\subseteq G\) with compact \(\overline{\psi(C)}\), then \(\psi\) is continuous.
Reviewer: Dieter Remus (Hagen)Tukey order and diversity of free abelian topological groupshttps://www.zbmath.org/1483.220032022-05-16T20:40:13.078697Z"Gartside, Paul"https://www.zbmath.org/authors/?q=ai:gartside.paul-mSummary: For a Tychonoff space \(X\) the \textit{free abelian topological group} over \(X\), denoted \(A(X)\), is the free abelian group on the set \(X\) with the coarsest topology so that for any continuous map of \(X\) into an abelian topological group its canonical extension to a homomorphism on \(A(X)\) is continuous.
We show there is a family \(\mathcal{A}\) of maximal size, \(2^{\mathfrak{c}}\), consisting of separable metrizable spaces, such that if \(M\) and \(N\) are distinct members of \(\mathcal{A}\) then \(A(M)\) and \(A(N)\) are not topologically isomorphic (moreover, \(A(M)\) neither embeds topologically in \(A(N)\) nor is an open image of \(A(N))\). We show there is a chain \(\mathcal{C}=\{M_\alpha:\alpha<\mathfrak{c}^+\}\), of maximal size, of separable metrizable spaces such that if \(\beta < \alpha\) then \(A( M_\beta)\) embeds as a closed subgroup of \(A( M_\alpha)\) but no subspace of \(A( M_\beta)\) is homeomorphic to \(A( M_\alpha)\).
We show that the character (minimal size of a local base at 0) of \(A(M)\) is \(\mathfrak{d}\) (minimal size of a cofinal set in \(\mathbb{N}^{\mathbb{N}})\) for every non-discrete, analytic \(M\), but consistently there is a co-analytic \(M\) such that the character of \(A(M)\) is strictly above \(\mathfrak{d}\).
The main tool used for these results is the Tukey order on the neighborhood filter at 0 in an \(A(X)\), and a connection with the family of compact subsets of an auxiliary space.A new fuzzy McShane integrabilityhttps://www.zbmath.org/1483.280152022-05-16T20:40:13.078697Z"Sayyad, Redouane"https://www.zbmath.org/authors/?q=ai:sayyad.redouaneSummary: We introduce the notion of the fuzzy McShane integral in the linear topology sense and we discuse its relation with the fuzzy Pettis integral introduced recently by \textit{C.-K. Park} in [Commun. Korean Math. Soc. 22, No. 4, 535--545 (2007; Zbl 1168.28314)].The stratic defuzzifier for discretised general type-2 fuzzy setshttps://www.zbmath.org/1483.684032022-05-16T20:40:13.078697Z"Greenfield, Sarah"https://www.zbmath.org/authors/?q=ai:greenfield.sarah"Chiclana, Francisco"https://www.zbmath.org/authors/?q=ai:chiclana.franciscoSummary: Stratification is a feature of the type-reduced set of the general type-2 fuzzy set, from which a new technique for general type-2 defuzzification, Stratic Defuzzification, may be derived. Existing defuzzification strategies are summarised. The stratified structure is described, after which the Stratic Defuzzifier is presented and contrasted experimentally for accuracy and efficiency with both the Exhaustive Method of Defuzzification (to benchmark accuracy) and the \(\alpha \)-Planes/Karnik-Mendel Iterative Procedure strategy, employing 5, 11, 21, 51 and 101 \(\alpha\)-planes. The Stratic Defuzzifier is shown to be much faster than the Exhaustive Defuzzifier. In fact the Stratic Defuzzifier and the \(\alpha\)-Planes/Karnik-Mendel Iterative Procedure Method are comparably speedy; the speed of execution correlates with the number of planes participating in the defuzzification process. The accuracy of the Stratic Defuzzifier is shown to be excellent. It is demonstrated to be more accurate than the \(\alpha \)-Planes/Karnik-Mendel Iterative Procedure Method in four of six test cases, regardless of the number of \(\alpha\)-planes employed. In one test case, it is less accurate than the \(\alpha\)-Planes/Karnik-Mendel Iterative Procedure Method, regardless of the number of \(\alpha\)-planes employed. In the remaining test case, the \(\alpha\)-Planes/Karnik-Mendel Iterative Procedure Method with 11 \(\alpha \)-Planes gives the most accurate result, with the Stratic Defuzzifier coming second.Some properties of membership functions composed of triangle functions and piecewise linear functionshttps://www.zbmath.org/1483.684912022-05-16T20:40:13.078697Z"Mitsuishi, Takashi"https://www.zbmath.org/authors/?q=ai:mitsuishi.takashiSummary: IF-THEN rules in fuzzy inference is composed of multiple fuzzy sets (membership functions). IF-THEN rules can therefore be considered as a pair of membership functions [\textit{E. H. Mamdani}, ``Application of fuzzy algorithms for control of simple dynamic plant'', IEE Proc. 121, No. 12, 1585--1588 (1974; \url{doi:10.1049/piee.1974.0328})]. The evaluation function of fuzzy control is composite function with fuzzy approximate reasoning and is functional on the set of membership functions. We obtained continuity of the evaluation function and compactness of the set of membership functions [\textit{T. Mitsuishi} et al., ``Continuity of defuzzification on \(L^2\) space for optimization of fuzzy control'', Lect. Notes Comput. Sci. 7669, 73--81 (2012; \url{doi:10.1007/978-3-642-35236-2_8})]. Therefore, we proved the existence of pair of membership functions, which maximizes (minimizes) evaluation function and is considered IF-THEN rules, in the set of membership functions by using extreme value theorem. The set of membership functions (fuzzy sets) is defined in this article to verifier our proofs before by Mizar [\textit{T. Mitsuishi}, \textit{N. Endou} and \textit{Y. Shidama}, ``The concept of fuzzy set and membership function and basic properties of fuzzy set operation'', Formaliz. Math. 9, No. 2, 351--356 (2001); \textit{T. Mitsuishi}, \textit{K. Wasaki} and\textit{Y. Shidama}, ``Basic properties of fuzzy set operation and membership function'', ibid. 9, No. 2, 357--362 (2001); \textit{A. Grabowski} and \textit{T. Mitsuishi}, Lect. Notes Comput. Sci. 9119, 160--171 (2015; Zbl 06595226)]. Membership functions composed of triangle function, piecewise linear function and Gaussian function used in practice are formalized using existing functions.
On the other hand, not only curve membership functions mentioned above but also membership functions composed of straight lines (piecewise linear function) like triangular and trapezoidal functions are formalized. Moreover, different from the definition in [\textit{A. Grabowski}, Formaliz. Math. 22, No. 4, 321--327 (2014; Zbl 1316.03030)] formalizations of triangular and trapezoidal function composed of two straight lines, minimum function and maximum functions are proposed. We prove, using the Mizar [\textit{G. Bancerek} et al., Lect. Notes Comput. Sci. 9150, 261--279 (2015; Zbl 1417.68201); J. Autom. Reasoning 61, No. 1--4, 9--32 (2018; Zbl 1433.68530)] formalism, some properties of membership functions such as continuity and periodicity [\textit{T. Mitsuishi}, ``Continuity of approximate reasoning using fuzzy number under Łukasiewicz t-norm'', in: 2015 IEEE 7th international conference on cybernetics and intelligent systems (CIS) and IEEE conference on robotics, automation and mechatronics (RAM). Los Alamitos, CA: IEEE Computer Society. 71--74 (2015; \url{doi:10.1109/ICCIS.2015.7274550}); \textit{T. Mitsuishi}, ``Uncertain defuzzified value of periodic membership function'', in: 2018 international electrical engineering congress (iEECON). Los Alamitos, CA: IEEE Computer Society. 1--4 (2018; \url{doi:10.1109/IEECON.2018.8712319})].QNMs of branes, BHs and fuzzballs from quantum SW geometrieshttps://www.zbmath.org/1483.830352022-05-16T20:40:13.078697Z"Bianchi, Massimo"https://www.zbmath.org/authors/?q=ai:bianchi.massimo"Consoli, Dario"https://www.zbmath.org/authors/?q=ai:consoli.dario"Grillo, Alfredo"https://www.zbmath.org/authors/?q=ai:grillo.alfredo"Morales, Francisco"https://www.zbmath.org/authors/?q=ai:morales.franciscoSummary: QNMs govern the linear response to perturbations of BHs, D-branes and fuzzballs and the gravitational wave signals in the ring-down phase of binary mergers. A remarkable connection between QNMs of neutral BHs in 4d and quantum SW geometries describing the dynamics of \(\mathcal{N} = 2\) SYM theories has been recently put forward. We extend the gauge/gravity dictionary to a large class of gravity backgrounds including charged and rotating BHs of Einstein-Maxwell theory in \(d = 4\), 5 dimensions, D3-branes, D1D5 `circular' fuzzballs and smooth horizonless geometries; all related to \(\mathcal{N} = 2\) SYM with a single \(SU(2)\) gauge group and fundamental matter. We find that photon-spheres, a common feature of all examples, are associated to degenerations of the classical elliptic SW geometry whereby a cycle pinches to zero size. Quantum effects resolve the singular geometry and lead to a spectrum of quantized energies, labelled by the overtone number \(n\). We compute the spectrum of QNMs using exact WKB quantization, geodetic motion and numerical simulations and show excellent agreement between the three methods. We explicitly illustrate our findings for the case D3-brane QNMs.Comparative analysis of fuzzy critical path method in agriculture project managementhttps://www.zbmath.org/1483.900662022-05-16T20:40:13.078697Z"Revathi, M."https://www.zbmath.org/authors/?q=ai:revathi.m"Valliathal, M."https://www.zbmath.org/authors/?q=ai:valliathal.mSummary: Agriculture is considered as a system that supplies valuable product and indefinite yields. It is essential to choose a suitable technique to maximize the yield and minimize losses in harvesting. Project management has techniques to assign activities that facilitate agriculture to standardize the quality, reduce expenses, develops the effectiveness and completes the project without delay. In this paper, an agriculture project has been constructed to minimize the cost of the project with the help of Fuzzy Critical Path Method (FCPM). Fuzzy numbers are more effective to deal with uncertainty which arises in the field of agriculture. Activities affecting the growth of saplings are identified, and a project network is drawn. The cost and duration of each activity is obtained from the observation. The expected duration for the completion of the project, total cost and the critical path is determined using a fuzzy network with generalized Trapezoidal fuzzy numbers, Heptagonal fuzzy number and Hendecagonal fuzzy number. The comparative analysis is done for a significant result of the fuzzy networking problem in agriculture project management.A decision-making approach to reduce the margin of error of decision makers for bipolar soft set theoryhttps://www.zbmath.org/1483.910632022-05-16T20:40:13.078697Z"Dalkılıç, Orhan"https://www.zbmath.org/authors/?q=ai:dalkilic.orhanSummary: In order for a mathematical model to express the uncertainty problems encountered in the most ideal way, it must be able to express the relationships between the parameters and objects in the problem in the most accurate way. In this paper, the bipolar soft set theory is taken into consideration since it also deals with the negative parameters of parameters in a parameter set. The main purpose of the paper is to determine the membership degrees between parameters and objects by minimising the effectiveness of the decision maker, and thus to build an impressive decision-making approach. For this, the concepts `bipolar relational membership function' and `NOT bipolar relational membership function' are defined and some important properties are given. Thanks to these proposed concepts, it is ensured that the decision maker is asked to express only firm judgments as 0 and 1, and the membership degrees between \((0,1)\) can be determined. Finally, the difference of our proposed decision-making approach from other decision-making approaches previously introduced in the literature has been clearly demonstrated and a comparative analysis has been made.On relative fuzzy soft sets over some semigroups in decision-making problemshttps://www.zbmath.org/1483.910672022-05-16T20:40:13.078697Z"Suebsan, Peerapong"https://www.zbmath.org/authors/?q=ai:suebsan.peerapongSummary: In this paper, we define relative fuzzy soft sets over some semigroups and give some their properties. Moreover, we construct a new algorithm for solving some decision-making problems based on relative fuzzy soft sets over some semigroups.Multiple criteria group decision-making method based on linguistic value soft fuzzy rough sets and its applicationhttps://www.zbmath.org/1483.910682022-05-16T20:40:13.078697Z"Zhong, Jiaming"https://www.zbmath.org/authors/?q=ai:zhong.jiaming"Zhang, Jialu"https://www.zbmath.org/authors/?q=ai:zhang.jialu"He, Minyuan"https://www.zbmath.org/authors/?q=ai:he.minyuanSummary: We study an application combining fuzzy sets, rough sets, soft sets, and linguistic valued variables in multiple criteria group decision-making. First, some operations on linguistic valued term sets are introduced, then a linguistic valued logical algebra system is established. The conclusion that a linguistic valued logical algebra system is an MV-algebra is also proved. A linguistic valued soft fuzzy rough approximation model is proposed, and its properties are discussed. Second, using a weighted average operator, the linguistic evaluation information of all experts for each decision project is aggregated, and a centralized evaluation for every decision from every criterion by the expert group is obtained. All centralized evaluation values constitute a linguistic soft set over the set of all decision projects. Third, an optimization model is established for computing the optimal criteria weight. The linguistic evaluation results of all experts for each criterion are aggregated using the optimal criteria weights, and then the evaluation values of all decision projects are obtained. Fourth, the evaluation values can be regarded as an linguistic valued soft set over the group of experts, where the parameter set is the set of all decision projects. Based on the linguistic valued soft fuzzy set over the set of decision projects, a rough approximation model is established. By computing the lower approximation and upper approximation of the evaluation value of all decision projects given by every expert, the lower approximation soft set and upper approximation soft set with linguistic valued over group experts are then formed. The weighted arithmetic mean of the evaluation sets of all decisions and their linguistic rough lower and upper approximations are used to obtain three linguistic valued fuzzy sets. By weighting the three linguistic valued fuzzy sets, the final evaluation of all decision projects is obtained and can be used to sort them or select the best one. Finally, the effectiveness and rationality of the presented method is verified by the case study of the multiple criteria comprehensive evaluation of online sales platforms for agricultural products.A new approach to fuzzy sets: application to the design of nonlinear time series, symmetry-breaking patterns, and non-sinusoidal limit-cycle oscillationshttps://www.zbmath.org/1483.940782022-05-16T20:40:13.078697Z"García-Morales, Vladimir"https://www.zbmath.org/authors/?q=ai:garcia-morales.vladimirSummary: It is shown that characteristic functions of sets can be made fuzzy by means of the \(\mathcal{B}_\kappa\)-function, recently introduced by the author, where the fuzziness parameter \(\kappa\in\mathbb{R}\) controls how much a fuzzy set deviates from the crisp set obtained in the limit \(\kappa\rightarrow 0\). As applications, we present first a general expression for a switching function that may be of interest in electrical engineering and in the design of nonlinear time series. We then introduce another general expression that allows wallpaper and frieze patterns for every possible planar symmetry group (besides patterns typical of quasicrystals) to be designed. We show how the fuzziness parameter \(\kappa\) plays an analogous role to temperature in physical applications and may be used to break the symmetry of spatial patterns. As a further, important application, we establish a theorem on the shaping of limit cycle oscillations far from bifurcations in smooth deterministic nonlinear dynamical systems governed by differential equations. Following this application, we briefly discuss a generalization of the Stuart-Landau equation to non-sinusoidal oscillators.