Katarzyna Troczka-Pawelec CONTINUITY OF
... real. Let Y be a topological vector space. Let n(Y ) denotes the family of all non-empty subsets of Y , c(Y )-the family of all compact members of n(Y ), cl(Y )-the family of all closed members of n(Y ) and Bcl(Y )-the family of all bounded and closed sets from n(Y ). The term set-valued function wi ...
... real. Let Y be a topological vector space. Let n(Y ) denotes the family of all non-empty subsets of Y , c(Y )-the family of all compact members of n(Y ), cl(Y )-the family of all closed members of n(Y ) and Bcl(Y )-the family of all bounded and closed sets from n(Y ). The term set-valued function wi ...
The Least Prime Number in a Beatty Sequence
... B(α, β) does not contain an entire residue class. It follows from a classical exponential sum estimate due to Ivan M. Vinogradov [17] that there exist infinitely many prime numbers in such a Beatty sequence (details in § 2), hence, in particular there exists a least prime number. However, the problem ...
... B(α, β) does not contain an entire residue class. It follows from a classical exponential sum estimate due to Ivan M. Vinogradov [17] that there exist infinitely many prime numbers in such a Beatty sequence (details in § 2), hence, in particular there exists a least prime number. However, the problem ...
1 Vector Spaces
... Definition 3.7 (Characteristic Polynomial). Given a square matrix A, we define the characteristic polynomial of A, p(x), as det(A − xI) Theorem 3.5. The Following Are Equivalent 1. λ is an eigenvalue of T 2. (∃x 6= 0) such that T x = λx 3. (∃x 6= 0) such that (T − λI)x = 0 4. T − λI is singular 5. N ...
... Definition 3.7 (Characteristic Polynomial). Given a square matrix A, we define the characteristic polynomial of A, p(x), as det(A − xI) Theorem 3.5. The Following Are Equivalent 1. λ is an eigenvalue of T 2. (∃x 6= 0) such that T x = λx 3. (∃x 6= 0) such that (T − λI)x = 0 4. T − λI is singular 5. N ...
On Some Aspects of the Differential Operator
... functional analysis course it is mentioned as an unbounded linear operator from the space C1[0,1] to C[0,1] under the sup or the uniform norm. When the Closed Graph Theorem is introduced, the differential operator serves as a counter example which asserts that although it is a closed operator, it is ...
... functional analysis course it is mentioned as an unbounded linear operator from the space C1[0,1] to C[0,1] under the sup or the uniform norm. When the Closed Graph Theorem is introduced, the differential operator serves as a counter example which asserts that although it is a closed operator, it is ...
On locally compact totally disconnected Abelian groups and their
... since a divisible subgroup of a discrete group is a direct factor. Hence ZP^((ZP, g), g)^(p~xgι, g)^{v°°, g)x(9u 9) which is a contradiction as observed above. Hence every neighborhood of g must contain a compact nonopen subgroup. This theorem shows that a reasonable conjecture for a possible auxili ...
... since a divisible subgroup of a discrete group is a direct factor. Hence ZP^((ZP, g), g)^(p~xgι, g)^{v°°, g)x(9u 9) which is a contradiction as observed above. Hence every neighborhood of g must contain a compact nonopen subgroup. This theorem shows that a reasonable conjecture for a possible auxili ...
Curious and Exotic Identities for Bernoulli Numbers
... property of involving Bernoulli sums both of type Br Bnr and r i.e., sums related to both the generating functions (A.1) and (A.2). In Sect. A.4 we look at products of Bernoulli numbers and Bernoulli polynomials in more detail. In particular, we prove the result (discovered by Nielsen) that when a ...
... property of involving Bernoulli sums both of type Br Bnr and r i.e., sums related to both the generating functions (A.1) and (A.2). In Sect. A.4 we look at products of Bernoulli numbers and Bernoulli polynomials in more detail. In particular, we prove the result (discovered by Nielsen) that when a ...
