Some notes on compact sets in soft metric spaces
... proved basic properties concerning soft topological spaces. Later, many researches about soft topological spaces were studied in [2, 7, 14, 19]. Soft mapping space and soft uniform space were studied in [11, 12]. Zahedi et al. studied fuzzy soft product topology and fuzzy soft boundary in [17, 18]. ...
... proved basic properties concerning soft topological spaces. Later, many researches about soft topological spaces were studied in [2, 7, 14, 19]. Soft mapping space and soft uniform space were studied in [11, 12]. Zahedi et al. studied fuzzy soft product topology and fuzzy soft boundary in [17, 18]. ...
Applied Science 174: Linear Algebra Lecture Notes
... For any α in R and any element (x1 , x2 , · · · , xn ) of Rn , we define: α · (x1 , x2 , · · · , xn ) = (αx1 , αx2 , · · · , αxn ). It is easy to verify that endowed with these two operations, Rn satisfies all the axioms of a real vector space. We can write therefore that (Rn , +, ·) is a real vecto ...
... For any α in R and any element (x1 , x2 , · · · , xn ) of Rn , we define: α · (x1 , x2 , · · · , xn ) = (αx1 , αx2 , · · · , αxn ). It is easy to verify that endowed with these two operations, Rn satisfies all the axioms of a real vector space. We can write therefore that (Rn , +, ·) is a real vecto ...
TOPOLOGY WITHOUT TEARS
... makes the study of topology relevant to all who aspire to be mathematicians whether their first love is (or will be) algebra, analysis, category theory, chaos, continuum mechanics, dynamics, geometry, industrial mathematics, mathematical biology, mathematical economics, mathematical finance, mathema ...
... makes the study of topology relevant to all who aspire to be mathematicians whether their first love is (or will be) algebra, analysis, category theory, chaos, continuum mechanics, dynamics, geometry, industrial mathematics, mathematical biology, mathematical economics, mathematical finance, mathema ...
Lesson 2: Linear and Non-Linear Expressions in
... into two parts, the left side and the right side. The two sides are called expressions. ...
... into two parts, the left side and the right side. The two sides are called expressions. ...
Complex varieties and the analytic topology
... setting, they didn’t have to worry about the Zariski topology and its many pathologies, because they already had a better-behaved topology to work with: the analytic topology inherited from the usual topology on the complex numbers themselves. In this note, we introduce the analytic topology, and ex ...
... setting, they didn’t have to worry about the Zariski topology and its many pathologies, because they already had a better-behaved topology to work with: the analytic topology inherited from the usual topology on the complex numbers themselves. In this note, we introduce the analytic topology, and ex ...
Companion to Real Analysis - Portland State University
... 1.1.2. CAUTION. In attempting to prove a theorem which has as a hypothesis “Let S be a set” do not include in your proof something like “Suppose S = {s1 , s2 , . . . , sn }” or “Suppose S = {s1 , s2 , . . . }”. In the first case you are tacitly assuming that S is finite and in the second that it is ...
... 1.1.2. CAUTION. In attempting to prove a theorem which has as a hypothesis “Let S be a set” do not include in your proof something like “Suppose S = {s1 , s2 , . . . , sn }” or “Suppose S = {s1 , s2 , . . . }”. In the first case you are tacitly assuming that S is finite and in the second that it is ...
LINEAR ALGEBRA Contents 1. Systems of linear equations 1 1.1
... at the origin point and head at any point in these spaces. The origin of the word “space” in the term “vector space” originates precisely in these first examples, which were associated with the physical space. Two operations are defined on oriented segments: An oriented segment can be stretched or c ...
... at the origin point and head at any point in these spaces. The origin of the word “space” in the term “vector space” originates precisely in these first examples, which were associated with the physical space. Two operations are defined on oriented segments: An oriented segment can be stretched or c ...
The Closed Limit Point Compactness
... closed limit point compact need not be a closed limit point compact. Also, we have shown that the quotient space of a closed limit point compact need not be a closed limit point compact. Finally, we have shown that if × is a closed limit point compact and is a -space, then is a closed limit point co ...
... closed limit point compact need not be a closed limit point compact. Also, we have shown that the quotient space of a closed limit point compact need not be a closed limit point compact. Finally, we have shown that if × is a closed limit point compact and is a -space, then is a closed limit point co ...
Metric Spaces
... defined on a subset of the real line, or the plane, or ordinary 3-space or, more generally, n-dimensional Euclidean space. The real line, plane, etc. are special cases of the general concept of “metric space’’ which is introduced in this chapter. We also introduce a convenient geometric language for ...
... defined on a subset of the real line, or the plane, or ordinary 3-space or, more generally, n-dimensional Euclidean space. The real line, plane, etc. are special cases of the general concept of “metric space’’ which is introduced in this chapter. We also introduce a convenient geometric language for ...
ON THE STRONG LAW OF LARGE NUMBERS FOR SEQUENCES
... are Theorems 3.1, 3.2, and 3.3. In Theorem 3.1 the random elements {li;" n ~ 1} are assumed to be blockwise independent with Eli;, = 0, n ~ 1, whereas in Theorem 3.3 the random elements are assumed to be blockwise p-orthogonal. In Theorem 3.2, it is shown that the implication (3.2) ~ (3.3) in Theore ...
... are Theorems 3.1, 3.2, and 3.3. In Theorem 3.1 the random elements {li;" n ~ 1} are assumed to be blockwise independent with Eli;, = 0, n ~ 1, whereas in Theorem 3.3 the random elements are assumed to be blockwise p-orthogonal. In Theorem 3.2, it is shown that the implication (3.2) ~ (3.3) in Theore ...
Projections in n-Dimensional Euclidean Space to Each Coordinates
... (19) Let a, b be real numbers, f be a map from ETn into R1 , and given i. Suppose that for every element p of the carrier of ETn holds f (p) = Proj(p, i). Then f −1 ({s : a < s ∧ s < b}) = {p; p ranges over elements of the carrier of ETn : a < Proj(p, i) ∧ Proj(p, i) < b}. (20) Let M be a metric spa ...
... (19) Let a, b be real numbers, f be a map from ETn into R1 , and given i. Suppose that for every element p of the carrier of ETn holds f (p) = Proj(p, i). Then f −1 ({s : a < s ∧ s < b}) = {p; p ranges over elements of the carrier of ETn : a < Proj(p, i) ∧ Proj(p, i) < b}. (20) Let M be a metric spa ...
Calculating generalised image and discriminant Milnor numbers in
... hypersurface. Since F is stable outside the origin has an isolated singularity at the origin. (This includes the case where is non-singular!) Let f be the restriction of F to . Then :¼ fðÞ is the discriminant of F. We shall be interested in the change of the local topology of this space (and ...
... hypersurface. Since F is stable outside the origin has an isolated singularity at the origin. (This includes the case where is non-singular!) Let f be the restriction of F to . Then :¼ fðÞ is the discriminant of F. We shall be interested in the change of the local topology of this space (and ...
Notes on topology
... by de Morgan’s Laws. Therefore X − (G1 ∩ G2 ) is finite (since it is the union of two finite sets), and G1 ∩ G2 ∈ Xf c as required. The topology Xf c is called the finite complement topology on X. 2.1. The topology induced by a metric. Let (X, d) be a metric space. Recall that a subset G of X is sai ...
... by de Morgan’s Laws. Therefore X − (G1 ∩ G2 ) is finite (since it is the union of two finite sets), and G1 ∩ G2 ∈ Xf c as required. The topology Xf c is called the finite complement topology on X. 2.1. The topology induced by a metric. Let (X, d) be a metric space. Recall that a subset G of X is sai ...
algebraic topology - School of Mathematics, TIFR
... Example 2.4 Let G = Z/(m) where m is any integer ≥ 0. Set k̄ + ¯l = k + l. It is easy to check that this defines an operation which satisfies our axioms. G becomes thus an abelian group and is finite if m > 0. Example 2.5 The non-zero real numbers denoted by R∗ (resp. the non-zero rational numbers d ...
... Example 2.4 Let G = Z/(m) where m is any integer ≥ 0. Set k̄ + ¯l = k + l. It is easy to check that this defines an operation which satisfies our axioms. G becomes thus an abelian group and is finite if m > 0. Example 2.5 The non-zero real numbers denoted by R∗ (resp. the non-zero rational numbers d ...
Vector Spaces - University of Miami Physics
... 2 The set of real-valued functions of a real variable, defined on the domain [a ≤ x ≤ b]. Addition is defined pointwise. If f1 and f2 are functions, then the value of the function f1 + f2 at the point x is the number f1 (x) + f2 (x). That is, f1 + f2 = f3 means f3 (x) = f1 (x) + f2 (x). Similarly, m ...
... 2 The set of real-valued functions of a real variable, defined on the domain [a ≤ x ≤ b]. Addition is defined pointwise. If f1 and f2 are functions, then the value of the function f1 + f2 at the point x is the number f1 (x) + f2 (x). That is, f1 + f2 = f3 means f3 (x) = f1 (x) + f2 (x). Similarly, m ...
Full text
... three-dimensional Euclidean space as triple products of Fibonacci numbers. This was achieved as a development of two-dimensional ideas involving complex numbers, though the three-dimensional extension was devoid of any dependence on complex numbers. Here, we wish to enlarge these notions to more gen ...
... three-dimensional Euclidean space as triple products of Fibonacci numbers. This was achieved as a development of two-dimensional ideas involving complex numbers, though the three-dimensional extension was devoid of any dependence on complex numbers. Here, we wish to enlarge these notions to more gen ...
Linear dependence and independence (chapter. 4)
... • If V is any vector space then V = Span(V ). • Clearly, we can find smaller sets of vectors which span V . • This lecture we will use the notions of linear independence and linear dependence to find the smallest sets of vectors which span V . • It turns out that there are many “smallest sets” of ve ...
... • If V is any vector space then V = Span(V ). • Clearly, we can find smaller sets of vectors which span V . • This lecture we will use the notions of linear independence and linear dependence to find the smallest sets of vectors which span V . • It turns out that there are many “smallest sets” of ve ...
P-adic Properties of Time in the Bernoulli Map
... sums Qp is the field of p-adic numbers, and contains the field of rational numbers Q but is different from it. ...
... sums Qp is the field of p-adic numbers, and contains the field of rational numbers Q but is different from it. ...
the linear vs. non
... x or y can NOT appear in the denominator. Both must be in the numerator!! x and y can NOT be multiplied together (no mushing). x or y can NOT have an exponent other than a 1 or a 0. ...
... x or y can NOT appear in the denominator. Both must be in the numerator!! x and y can NOT be multiplied together (no mushing). x or y can NOT have an exponent other than a 1 or a 0. ...
Linear and Nonlinear Functions
... x or y can NOT appear in the denominator. Both must be in the numerator!! x and y can NOT be multiplied together (no mushing). x or y can NOT have an exponent other than a 1 or a 0. ...
... x or y can NOT appear in the denominator. Both must be in the numerator!! x and y can NOT be multiplied together (no mushing). x or y can NOT have an exponent other than a 1 or a 0. ...
Linear and Nonlinear Functions
... x or y can NOT appear in the denominator. Both must be in the numerator!! x and y can NOT be multiplied together (no mushing). x or y can NOT have an exponent other than a 1 or a 0. ...
... x or y can NOT appear in the denominator. Both must be in the numerator!! x and y can NOT be multiplied together (no mushing). x or y can NOT have an exponent other than a 1 or a 0. ...
The Uniform Continuity of Functions on Normed Linear Spaces
... (ii) for every r such that 0 < r there exists s such that 0 < s and for all x1 , x2 such that x1 ∈ X and x2 ∈ X and kx1 − x2 k < s holds |fx1 − fx2 | < r. The following propositions are true: (1) If f is uniformly continuous on X and X1 ⊆ X, then f is uniformly continuous on X1 . (2) If f1 is unifor ...
... (ii) for every r such that 0 < r there exists s such that 0 < s and for all x1 , x2 such that x1 ∈ X and x2 ∈ X and kx1 − x2 k < s holds |fx1 − fx2 | < r. The following propositions are true: (1) If f is uniformly continuous on X and X1 ⊆ X, then f is uniformly continuous on X1 . (2) If f1 is unifor ...
On some definition of expectation of random element in
... give natural convex combination as is written in [14]. The property (ii) may not be satisfied if we define the convex combination operation [p1 , xi ]ni=1 as an expectation of random element taking values xi with probability pi respectively. The reason of this fact is that not for all definitions of ...
... give natural convex combination as is written in [14]. The property (ii) may not be satisfied if we define the convex combination operation [p1 , xi ]ni=1 as an expectation of random element taking values xi with probability pi respectively. The reason of this fact is that not for all definitions of ...