More conservativity for weak Kőnig’s lemma

The compact quotients of three-dimensional solvable non-nilpotent Lie groups by discrete subgroups fall into two categories; those which are quotients of $S\sb{R}$ (semidirect product, R acts on R$\sp2$ by rotation), and those which are quotients of $S\sb{H}$ = $R\ \propto\ R\sp2$ (semidirect product, $R$ acts on $R\sp2$ via translation along a hyperbolic orbit). Continuous functions in the primary summands of $L\sp2$ of each type of solvmanifold are examined. It is shown that there exists a Fejer theorem on solvmanifolds $M\sb{R}$ which are quotients of $S\sb{R}$; if $P\sb{i}$: $L\sp2(M\sb{R})\mapsto$ $L\sp2(M\sb{R})$ is orthogonal projection onto the ith primary summand of such a quotient, there exists a sequence of operators $S\sb{n}$ such that each $S\sb{n}$ is a finite linear combination of the $P\sb{i}$, and such that for each continuous function $f\in L\sp2(M\sb{R}),\ S\sb{n}(f)\to f$ uniformly as $n\to\infty$. It is shown that no such theorem holds for quotients of $S\sb{H}$, for which it is known that orthogonal projections do not preserve continuity of functions. It is shown that if $f\in L\sp2(M\sb{H})$ is continuous and $P\sb{i}(f)$ is essentially bounded, then $P\sb{i}(f)$ must be continuous. Together with a result of L. Richardson, this result implies that there are continuous $f\in L\sp2(M\sb{H})$ for which $P\sb{i}(f)$ is essentially unbounded. It is shown in Section 2 that continuous functions in primary summands of $L\sp2(M\sb{H})$ must vanish on $M\sb{H}$ for all compact quotients of $M\sb{H}$ by discrete subgroups. There are five compact quotients of $S\sb{R}$; one quotient manifold is homeomorphic to the three-dimensional torus, and its continuous primary summand functions need not vanish. For three of the other quotients, it is shown that all continuous primary summand functions need not vanish. For the remaining quotient it is known that functions in certain subspaces of the primary summands must vanish, but it is not known whether all continuous primary summand functions must vanish in this case.

In this paper, we continue to study the properties of the relation with some type of open sets, and we introduce -continuous function, semi-continuous function, -continuous function, and -continuous function are studied and some of their characteristics are discussed. In this work, we need to introduce the concepts of function, especially the inverse function to find all continuous function, so we want to prove some examples, theorems, and observations of our subject with the help of new concepts for the alpha-open sets of sums to make it easier for us to find a relationship between these formulas as well as the converse relationship has been studied and explained with illustration many examples. Hence, reaching to get a relationship (continuous, -continuous, semi -continuous) function at new condition.

A new class of closed sets, known as semi-regular weekly closed sets, is introduced in ideal topological space. In addition, we define a new class of continuous functions known as semi-regular weekly -continuous functions is introduced and studied. Some of their characteristics have also been examined and discussed interesting properties of semi-regular weekly -irresolute functions, strongly semi-regular weekly -continuous functions, and semi- regular weekly -continuous functions.

In this paper a new class of functions, such that semi –continuous function is introduced for topological spaces, and the second type is semi –continuous function and three class is semi –continuous function. We have taken in our study this continuous homeomorphism function (bijective = injective + surjective). we introduce using practical examples of mathematical formulas and considering them as a direct application to the validity of the observations. We also, study the relationship between these concepts that we referred to at the beginning of the research.

A space X is said to have the f.p.p. (fixed point property) if every continuous function f from X to X has a fixed point. Whether if X and Y have the f.p.p. then XX Y has the f.p.p. is an open question. A space X is said to have the F.p.p. (fixed point property for multivalued functions) if every continuous multi-valued function F from X to X has a fixed point, i.e., a point x such that xGF(x). Interest in fixed points for multi-valued functions leads one to question under what conditions on the spaces X and Y and on the multi-valued function F on X to Y there will exist a continuous trace f of F, that is, a continuous function f on X to Y such that f(x) E F(x) for all x. For some specific multi-valued functions F it is possible to produce a continuous trace. It is by use of these traces that most of the fixed point theorems in the literature for multi-valued functions are proved. In fact the open question mentioned above (which is concerned only with single-valued functions) can be answered if one can produce continuous traces of two particular multi-valued functions. This paper proves some fixed point theorems by producing continuous traces, shows that a continuous multi-valued function need not have a continuous trace, and gives an example which indicates that a general theorem on the existence of a continuous trace is not likely to be established without strong conditions on F regardless of what conditions are placed on X and Y. This example answers in the negative the generalization of the above open question to the multi-valued case, exhibits a continuous multi-valued function which has no continuous trace, shows that the general Tychonoff cube does not have the F.p.p., and shows that a space with the f.p.p. need not have the F.p.p. NOTATION. By {1x }-*xo we denote a sequence of points indexed by a directed set A and converging to xo. The directing relation in A will be denoted by *. DEFINITION 1. Continuous. A multi-valued function on a space X to a space Y is said to be continuous at xo if {Xa } -+x0 implies that F(xo) = cofinal limit { F(xa) } = residual limit { F(Xa) }. F is said to be

For a metric continuum $X$, let $C(X)$ (resp., $2^{X}$) be the hyperspace of subcontinua (resp., nonempty closed subsets) of $X$. Let $f:X\rightarrow Y$ be an almost continuous function. Let $C(f):C(X)\rightarrow C(Y)$ and $ 2^{f}:2^{X}\rightarrow 2^{Y}$

We appreciate Tihomir Ancev’s interest in our research on the economic consequences of multiyear droughts (Peck and Adams 2010). Ancev raises two concerns about our paper in their comment. First, he is concerned that our main conclusion (i.e. drought in one year can influence the effects of drought in subsequent years) may be an artefact of our representation of crop interdependencies as ‘agronomic rules.’ He asks, specifically, whether our conclusion would change if we allowed the model to make trade-offs between drought management and pest and disease management. Ancev’s second concern is that our conclusion does not hold for all drought scenarios we analysed. Because our conclusion does not hold universally (across all states of nature in the analysis), Ancev questions its validity. He closes by recommending an alternative approach that would compare the effects of single versus multiyear drought events. Ancev is correct that agricultural producers facing drought are not rigidly bound by agronomic rules, but instead face trade-offs between crops that help manage scarce water resources and those that help manage pests and diseases. Producers may violate agronomic rules if they are more concerned about mitigating the financial impact of drought in the current year than the financial impact of increased pests and diseases in future years. Continuous production functions would capture the yield effects of alternative crop sequences more precisely and allow marginal trade-offs between drought and pest/disease management. Unfortunately, we found no such production functions in the literature for our study area’s major crops (wheat, corn, sugar beets and onions). El-Nazer and McCarl (1986) estimated production functions for wheat and corn in the Pacific Northwest, but their study did not include beets or onions, so their production functions do not capture the yield effects of sugar beets or onions on wheat and corn, or vice versa. Nor were adequate production data available from the study area to estimate functions that capture crop interdependencies. Our use of agronomic rules is based on interviews with producers in the region. These producers indicated that agronomic rules about crop rotations are an important means for managing diseases and pests, maintaining soil quality, and minimising input costs. Producers also expressed a general belief that the costs of deviating from agronomic rules generally outweigh the benefits; that is, local agronomic rules already reflect economically optimal crop rotation practices under most growing conditions. No producers indicated that they deviate from agronomic rules as a drought management strategy. We believe the crop rotation rules used in our study accurately reflect farmer behaviour (in this region at least) and are preferable to more restrictive approaches, such as pre-determined crop rotations. However, the more important issue from the standpoint of Ancev’s comment is whether use of these agronomic rules restricts model solutions in ways that invalidate our core finding. Agronomic rules clearly provide less flexibility in the model’s crop choice than would continuous production functions. Consequently, net revenue arising from our model should be less (or no greater) than net revenue if we used continuous functions instead. It is evident, however, that agronomic rules capture the fundamental dilemma producers face given uncertain water supplies and interyear crop dynamics; decisions they make in response to drought in the current year have implications for future cropping opportunities and hence their vulnerability to future years of drought. As long as our model captures the farm system’s interyear crop dynamics (using either agronomic rules or continuous production functions), the result that drought can generate impacts beyond the years in which it occurs will hold. A simple example illustrates that our conclusion is not an artefact of agronomic rules and that it would also hold if continuous production functions were used. Suppose a producer violates one of the agronomic rules by planting wheat in the same field in two consecutive years. They might do so because they believe having a larger portion of their total crop acreage in a drought-tolerant crop such as wheat reduces the expected cost of drought. The benefit of this decision is the ability to manage scarce water resources more effectively. However, the decision’s cost is that consecutive wheat crops increase cereal cyst nematode populations to densities capable of reducing yield (Smiley and Yan 2010). Assume the producer decides, in the third year, to address the nematode problem by planting an alternative crop in the infested field. The chosen crop is presumably less drought-tolerant than wheat; otherwise, the producer would have planted it in the second year instead of wheat. If drought occurs in the third year, the producer experiences greater economic losses than if they had planted wheat, because the alternative crop requires more water and is more sensitive to deficit irrigation. Even if the producer’s gains in the second year outweigh their additional costs in the third year, the conclusion is the same: decisions made in preparation for, or in response to drought in the second year influenced the effects of drought in the third year. To complete this multiyear example, suppose the producer decides in the third year to instead plant wheat in the same field (for a third consecutive year). When drought occurs in the third year, wheat will be especially vulnerable to the burgeoning nematode population (Smiley and Yan 2010), and yield losses will be larger than if nematodes were not so abundant (i.e. than if the producer had not planted wheat in the second year). Regardless of which crop is planted in the third year, the producer’s decision to plant wheat in the second year (to reduce the expected cost of drought, at the expense of future pest management) influences the cost of drought in the third year. Replacing agronomic rules with continuous production functions would not change the main conclusion that drought preparedness and response activities (and the resulting financial impacts of drought) in one year can influence the effects of drought in subsequent years. This discussion highlights a need, nonetheless, for continuous production functions that describe interdependencies between a wider range of crops, and a more complete understanding of the short and long-term benefits and costs of deviating from typical crop rotations. Ancev’s second concern is that our conclusion does not hold for all drought scenarios analysed within our model. Table F1 in the supplementary appendix to Peck and Adams (2010) does reveal that drought in one year does not always influence the effects of drought in subsequent years. Specifically, in five of 64 water supply scenarios, drought in previous years does not influence the effects of drought in subsequent years. In these five scenarios, drought preparedness and response activities are sufficiently moderate that they do not impose binding constraints for cropping activities in subsequent years. If the initial crop plan has sufficiently low water requirements, only subtle adjustments (e.g. deficit irrigation) are necessary to mitigate water shortages. Deficit irrigation imposes no constraints on cropping activities in subsequent years and therefore would not alter the effects of a subsequent year of drought. In two of 64 scenarios, drought in one year actually reduces the effects of drought in a subsequent year. Again, characteristics of the crop plans in place when drought occurs dictate the extent of mitigation necessary to address a water shortage, and hence the extent to which current and future crop plans might have to be altered. If the initial crop plan’s water requirements are sufficiently high, dramatic mitigation measures may be necessary. Some mitigation measures actually improve a producer’s ability to mitigate drought in subsequent years. Suppose, for example, a producer fallows/abandons a wheat field in response to drought. This creates the option to re-plant the field to wheat the following year (without violation of agronomic rules), which would improve the producer’s future drought management capabilities. The conclusion that a multiyear drought can be more than the sum of its parts holds for 57 of 64 water supply scenarios and is therefore in our opinion quite robust. This finding provides evidence that the effects of an individual year of drought can depend on the occurrence of drought in previous years, although such interdependence is not guaranteed. We would have liked to have presented and discussed the few exceptions to our conclusion in our original paper, but lacked sufficient space to do so. Ancev concludes the comment by recommending an alternative approach for comparing the effects of single versus multiyear drought events. In response, we would like to clarify that our paper’s purpose is not to compare the effects of single versus multiyear drought events. It was not our intent to determine, for example, whether a 3-year drought generates larger impacts than a 2-year drought, or a 2-year drought generates larger impacts than a 1-year drought, although Table F1 can be used to answer such questions. Our purpose is instead to determine whether the effects of individual years of a multiyear drought event are independent of one another, i.e. whether the effects of a multiyear drought can be decomposed into independent annual components. The simulation procedure Ancev recommends in his final paragraph would not address this question. However, it would contribute other interesting insights about multiyear drought.

A new class of functions, called slightly generalized β-continuous functions is introduced.Basic properties of slightly generalized β-continuous functions are studied.The class of slightly generalized βcontinuous functions properly includes the class of slightly β-continuous functions and generalized β-continuous functions.Also, by using slightly generalized β-continuous functions, some properties of domain/range of functions are characterized.

Approximations of relations by continuous functions

Urysohn [4, p. 290] has considered the problem of determining the most general class of topological spaces in which non-constant real-valued continuous functions exist. He showed that a countable Hausdorff space exists in which every continuous real-valued function is constant [4, pp. 274-283]. Pospisil [2, pp. 1-3] has extended this result by exhibiting a Hausdorff space of arbitrary infinite cardinal number in which every continuous real-valued function is constant. The spaces constructed by both Urysohn and Posp ?il, however, fail to satisfy the Urysohn separation axiom.2 Urysohn posed, but left unanswered, the question of whether or not regular topological spaces exist in which every continuous real-valued function is constant. In the present paper, it will be proved that regular spaces exist in great profusion on which every real-valued continuous function is constant. Urysohn also considered the problem of determining the least cardinal number of a connected space. [4, pp. 274-283], and, by the construction mentioned above, showed that a countable connected Hausdorff space exists. This example, however, does not satisfy the Urysohn separation axiom. Urysohn's result is extended in the present paper by the construction of a countable -Urysohn space in which every continuous function is constant and which is accordingly connected. THEORBM 1. Let N be an infinite cardinal number which is not the sum of No cardinal numbers smaller than R. Then there exists a regular space 'k of cardinal number N in which every continuous real-valued function is constant. Our construction of the space ?k, which proceeds in several steps, makes essential use of a space constructed by Tychonoff [3, pp. 553-556] and the condensation process devised by Urysohn [4, pp. 274-283]. First, let A be the least ordinal number with corresponding cardinal number N. It is obvious that A is not the limit of a countable sequence of ordinal numbers smaller than A. Let D be the set of all ordinal numbers a such that 1 ? a < A Let C be the set of all ordinal numbers a such that 1 < a ? c, where X denotes, as usual, the least infinite ordinal number. D and C are made into topological spaces by the following construction. In D, every point 6, where 1 < a < A, is isolated. Neighborhoods Uy(A) of the point A e D are the sets E[b; y < _5 ?A]. C is given a similar neighborhood system with X as the only non-isolated point. Let S be the set of all pairs (6,a) of ordinal numbers such that ( E D and a E C.

A special class of approximations of measurable functions of several variables on the unit coordinate cube is investigated. The class is constructed on the base of Kolmogorov's theorem (in version by Sprecher–Golubkov) stating that a continuous function $f$ of several variables can be represented as a finite superposition of continuous single-variable functions — so called outer functions (which depend of $f$) and inner one $\Psi$ (which is independent of $f$ and is monotone). In the case of continuous functions $f$ the class under study is obtained with outer functions approximated by linear combinations of quadratic exponentials (also known as Gaussian functions or Gaussian kernels) and with the inner function $\Psi$ approximated by Laplace functions. As is known, a measurable function $f$ can be approximated by a continuous one (up to a set of small measure) with the help of classical Luzin's theorem. The effectiveness of such approach is based on assertions that, firstly, the Mexican hat mother wavelet on any fixed bounded interval can be approximated as accurately as desired by a linear combination of two Gaussian functions, and, secondly, that a continuous monotone function on such an interval can be approximated as accurately as desired by a linear combination of translations and dilations of the Laplace integral (in other words, Laplace functions). It is proved that the class of approximations under study is dense everywhere in the class of continuous multivariable functions on the coordinate cube. For the case of continuous and piecewise continuous functions of two variables, numerical results are presented that confirm the effectiveness of approximations of the studied class.

A unified theory of continuous and certain non-continuous functions is proposed and developed. The proposed theory encompasses in one the theories of continuous functions, upper (lower) semicontinuous functions, almost continuous functions, c-continuous functions, c*-continuous functions, s-continuous functions, l-continuous functions, H-continuous functions, and the ε-continuous functions of Klee.

Process Takagi–Sugeno model: A novel approach for handling continuous input and output functions and its application to time series prediction

A unified theory of continuous and certain non-continuous functions, initiated in an earlier paper, is further elaborated. The proposed theory provides a common platform for dealing simultaneously with continuous functions and a host of non-continuous functions including lower (upper) semicontinuous functions, almost continuous functions, weakly continuous functions (encountered in functional analysis), c-continuous functions, δ-continuous functions, semiconnected functions, H-continuous functions s-continuous functions, ε-continuous functions of Klee and several other variants of continuity.

Numerical relationships between neural networks, continuous functions, and fuzzy systems