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An Introduction to Mathematical Reasoning: Bibliography

An Introduction to Mathematical Reasoning, Dec 11, 1997

Definition 0.0.1 Fix n ∈ Z with n > 1. We say that two integers a and b are congruent modulo n an... more Definition 0.0.1 Fix n ∈ Z with n > 1. We say that two integers a and b are congruent modulo n and write a ≡ b (mod n) provided that n | (b − a).

The language of mathematics

An Introduction to Mathematical Reasoning

The sequence of prime numbers

Congruence modulo a prime

The division theorem

Let a and b be integers with b ≠ 0. Then there exists integers q and r such that: a = bq + r and ... more Let a and b be integers with b ≠ 0. Then there exists integers q and r such that: a = bq + r and 0 ≤ r ≤ | b | In fact the integers q and r are unique. The rule that b must be non-negative is convention and guarantees uniqueness.

Mathematical statements and proofs

We consider the problem of detecting Mahowald's family ηi ∈ 2π S 2 i in homology. This allows us ... more We consider the problem of detecting Mahowald's family ηi ∈ 2π S 2 i in homology. This allows us to identify specific spherical classes in H * Ω 2 i+1 −8+k 0 S 2 i −2 for 0 k 6. We then identify the type of the subalgebras that these classes give rise to, and calculate the A-module and R-module structure of these subalgebras. We shall the discuss the relation of these calculations to the Curtis conjecture on spherical classes in H * Q0S 0 .

Congruence classes and the arithmetic of remainders

Counting functions and subsets

Numbers and counting

Cambridge University Press, Dec 1, 1997

Linear diophantine equations

An Introduction to Mathematical Reasoning

Proceedings of the Steklov Institute of Mathematics, 2006

We present a geometrical version of Herbert's Theorem determining the homology classes represente... more We present a geometrical version of Herbert's Theorem determining the homology classes represented by the multiple point manifolds of a self-tranverse immersion. Herbert's Theorem and generalizations can readily be read off from this result. The simple geometrical proof is based on ideas in Herbert's paper. We also describe the relationship between this theorem and the homotopy theory of Thom spaces.

We consider the problem of calculating the Hurewicz image of Mahowald's family η_i∈_2π_2^i^S.... more We consider the problem of calculating the Hurewicz image of Mahowald's family η_i∈_2π_2^i^S. This allows us to identify specific spherical classes in H_*Ω_0^2^i+1-8+kS^2^i-2 for 0≤ k≤ 6. We then identify the type of the subalgebras that these classes give rise to, and calculate the A-module and R-module structure of these subalgebras. We shall the discuss the relation of these calculations to the Curtis conjecture on spherical classes in H_*Q_0S^0, and relations with spherical classes in H_*Q_0S^-n.

An Introduction to Mathematical Reasoning: Bibliography

An Introduction to Mathematical Reasoning, Dec 11, 1997

Definition 0.0.1 Fix n ∈ Z with n > 1. We say that two integers a and b are congruent modulo n an... more Definition 0.0.1 Fix n ∈ Z with n > 1. We say that two integers a and b are congruent modulo n and write a ≡ b (mod n) provided that n | (b − a).

The language of mathematics

An Introduction to Mathematical Reasoning

The sequence of prime numbers

Congruence modulo a prime

The division theorem

Let a and b be integers with b ≠ 0. Then there exists integers q and r such that: a = bq + r and ... more Let a and b be integers with b ≠ 0. Then there exists integers q and r such that: a = bq + r and 0 ≤ r ≤ | b | In fact the integers q and r are unique. The rule that b must be non-negative is convention and guarantees uniqueness.

Mathematical statements and proofs

We consider the problem of detecting Mahowald's family ηi ∈ 2π S 2 i in homology. This allows us ... more We consider the problem of detecting Mahowald's family ηi ∈ 2π S 2 i in homology. This allows us to identify specific spherical classes in H * Ω 2 i+1 −8+k 0 S 2 i −2 for 0 k 6. We then identify the type of the subalgebras that these classes give rise to, and calculate the A-module and R-module structure of these subalgebras. We shall the discuss the relation of these calculations to the Curtis conjecture on spherical classes in H * Q0S 0 .

Congruence classes and the arithmetic of remainders

Counting functions and subsets

Numbers and counting

Cambridge University Press, Dec 1, 1997

Linear diophantine equations

An Introduction to Mathematical Reasoning

Proceedings of the Steklov Institute of Mathematics, 2006

We present a geometrical version of Herbert's Theorem determining the homology classes represente... more We present a geometrical version of Herbert's Theorem determining the homology classes represented by the multiple point manifolds of a self-tranverse immersion. Herbert's Theorem and generalizations can readily be read off from this result. The simple geometrical proof is based on ideas in Herbert's paper. We also describe the relationship between this theorem and the homotopy theory of Thom spaces.

We consider the problem of calculating the Hurewicz image of Mahowald's family η_i∈_2π_2^i^S.... more We consider the problem of calculating the Hurewicz image of Mahowald's family η_i∈_2π_2^i^S. This allows us to identify specific spherical classes in H_*Ω_0^2^i+1-8+kS^2^i-2 for 0≤ k≤ 6. We then identify the type of the subalgebras that these classes give rise to, and calculate the A-module and R-module structure of these subalgebras. We shall the discuss the relation of these calculations to the Curtis conjecture on spherical classes in H_*Q_0S^0, and relations with spherical classes in H_*Q_0S^-n.

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