We consider purely singular homogeneous Young measures associated with elements of sequences of piecewise constant functions and with limits of such sequences. We first consider a case when the limit of a such sequence is piecewise constant. The next point involves the sequences of bounded oscillating functions, divergent in the strong topology in L ∞ , but weakly∗ convergent to a homogeneous Young measure. We also present an example of a fast oscillating sequence, illustrating the result. In the presented results, generalizing to some extent known examples, we try to avoid advanced methods of functional analysis that are usually used when solving problems of this type.
Artykuł jest kontynuacją artykułu „Obliczanie granic ciągów o wyrazach rzeczywistych”. Przedstawiono typowe przykłady obliczania granic funkcji rzeczywistych jednej zmiennej rzeczywistej. W związku z tym prezentowane przykłady są bardzo typowe, zawierają dokładne wyjaśnienia, co i dlaczego robimy. Na początku znajduje się lista twierdzeń oraz podstawowych granic, z których będziemy korzystać. Po każdym przykładzie znajdują się dwa bardzo podobne zadania do samodzielnego rozwiązania, na końcu kilka dodatkowych zadań.
The paper shows how to identify discontinuities of real valued functions of one real variable and how to determine types of discontinuities. There are a few practical tasks with step by step solutions.
We prove in this paper that if the composition operator H, generated by a function h : I b a x C(Iba) Y , maps ɸBV (Iba ,C) into ɸ2 BV (Iba , Y ) and is uniformly continuous, then the left-left regularization h* of h is an affine function with respect to the third variable.
Let S be an integral domain of positive characteristic p, which is not a field, S∗ the unit group of S, G a finite group, and SλG the twisted group algebra of the group G over S with a 2-cocycle λ ∈ Z2(G,S∗). Denote by Indm(SλG) the set of isomorphism classes of indecomposable SλGmodules of S-rank m. We exhibit algebras SλG of SUR-type, in the sense that there exists a function fλ : N → N such that fλ(n) ≥ n and Indfλ(n)(SλG) is an infinite set for every integer n > 1.
The concept of strong convergence of functions and multifunctions was introduced by I. Kupka, V. Toma and A. Sochaczewska. In this paper we consider new definitions of convergence for the nets of multifunctions – upper and lower strong quasi-uniform convergence.
In the present paper, a few different notions of preponderant continuity of a real function are discussed. We study the relationship between them and give some properties of preponderant continuity.
In this paper we present generalization of probability density of random variables. It is obvious that probability density is definite only for absolute continuous variables. However, in many practical applications we need to define the analogous concept also for variables of other types. It can be easily shown that we are able to generalize the concept of density using distributions, especially Dirac’s delta function.
The paper includes a necessary condition and sufficient conditions under which two ψ -sparse topologies generated by two functions ψ1 and ψ 2 are equal. Additionally we proved that the intersection of all ψ -sparse topologies is equal to the Hashimoto topology.
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Let (X, +) be an Abelian group. One can show that a mapping f: X R satisfying the inequality f(x + y) + f(x-y)≤2f(x)+2f(y) (1) for all x, y ∈ X also satisfies the inequalities f(2x + y)≤4f(:c) + f (y) + f (x + y) - f (x - y) and f(2x+y) + f(2x -y)≤ 8f(x) + 2f(y) for all x, y ∈ X. A question of finding sufficient conditions under which the inequalities (1), (2) and (3) are equivalent will be considered. In this note, some properties of the solution of (1) will be proved. We also consider another definition of a subquadratic function given in [1].
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We consider some properties of functions defined in a topological space X with values in another topological space Y. We shall consider some properties of functions defined in a topological space X with values in a topological space Y. Topological terminology is taken from the books General Topology by R. Engelking [1] and General Topolowy by J. L. Kelley [7].
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The notion of function plays a crucial role in teaching mathematics. Why is this important issue so problematic for students? It is worth noticing that precise specification of this notion took place relatively late, in 19th century. This is why it is so important to attempt students” active participation in defining and understanding the notion of „function”.
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Periodic B-spline functions have got many useful properties. Especially it is the property of its Fourier coefficients. In this article it is introduced and proved a similar property of Fourier coefficients of spline wavelets.
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lf (X, ᵨ) is a dense in itself metric space and f : X →ℝ, then we define ω*(f,x) = infr >0 supy,z∈B (x,r) \ {x} ׀ f(y) - f(z)׀. We say that a function F : X →ℝ is an ω*-primitive for f : X →ℝ if ω* (F, .) = f. We discuss problem of the existence of ω*-primitives for an arbitrary upper semicontinuous function f : X → [0, ∞ ) defined on a dense in itself metric space. At the end we show that if an upper semicontinuous function f : X → [0, ∞) is defined on a nonmetrizable topological space, then ω*-primitive may not exists.
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Many students have problems with solving tasks concerning the existence of the derivative of a function at a point. In this paper we discuss some of them.
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In this paper we consider various definitions of a periodic function and establish connections between them, in particular, we prove equivalence of some of them. In papers and textbooks one can find different definitions of a periodic function. This raises the question which of them are equivalent.
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In this paper we consider various definitions of a periodic function and establish connections between them, in particular, we prove equivalence of some of them. In papers and textbooks one can find different definitions of a periodic function. This raises the question which of them are equivalent.
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