## machine learning – How to prove the lemma of Nagarajan?

Let $$H$$ be a class of multi-class predictor hypotheses; namely, each $$h H$$ is a function of $$X$$ at $$(k)$$.

Note the Natarajan dimension of $$H$$ through $$Ndim (H)$$. Please prove the following lemma.

$$| H | le | X | ^ {Ndim (H)} cdot k ^ {2Ndim (H)}$$

The lemma is in the book "Understanding machine learning: from theory to algorithms". You have just searched for keywords "Lemma 29.4".

## Elementary Number Theory – Prove with Mathematical Induction

Show that for each natural number n, with n> = 3, (1+ (1 / n) ^ n <n.

I've taken the basic step, (4/3) ^ 3 <3 so that's fine, but I'm confused about the inductive step.

The inductive hypothesis is (1+ (1 / k) ^ k <k, and the inductive conclusion is (1+ (1 / (k + 1))) ^ k + 1 <k + 1. I n & # 39 can not understand how to reach the IC since the IH.

I've tried to multiply both sides by (1+ (1 / k), but that just gives (1+ (1 / k)) ^ k + 1 <k + 1. Idk what I'm supposed to do right here?

## General Topology – Prove that the function is continuous

Let (X,$$μ_1$$), (X,$$μ_2$$), (Y,$$Ω_1$$), (Y, $$Ω_2$$) Topological spaces.
Let $$μ_1$$ , $$Ω_1$$ to be finer than $$μ_2$$ ,$$Ω_2$$ (that means: $$μ_1$$ $$subset$$ $$μ_2$$ , same $$Ω_1$$ , $$Ω_2$$. Suppose that f: (X,$$μ_1$$) -> (Y,$$Ω_2$$) is continuous.
Prove that: 1. g: (X,$$μ_2$$) -> (Y,$$Ω_2$$) is a continuous function.
2. h: (X,$$μ_1$$) -> (Y,$$Ω_1$$) is a continuous function.
In 1 I said: f is continuous, so according to the definition, for every U $$in$$ Ω_2, f ^ {- 1} (U) $$in$$ μ_1.
We know that $$μ_1$$ $$in$$ $$μ_2$$ then f ^ {- 1} (U) $$in$$ $$μ_2$$ . So, according to the definition of a continuous function, we get that g is continuous.
Is it correct?
In 2, I did not manage to make the link between the definitions, 1 and what is given.

## complex numbers – Prove that \$ {(i ^ {2n} +1) (i ^ n + 2): n in mathbb {N} } = {0,2,6 } \$ and that \$ i ^ m \$ is \$ 4 \$ -periodic

$$1.$$ Prove it $${(i ^ {2n} +1) (i ^ n + 2): n in mathbb {N} } = {0,2,6 }$$.

$$2.$$ Prove it $$forall m in mathbb {Z}, i ^ m = begin {case} 1, & text {if m equiv 0 pmod 4 } \ i, & text {if m equiv 1 pmod 4 } \ – 1, & text {if m equiv 2 pmod 4 } \ – i, & text {if m equiv 3 pmod 4 } \ end {cases}$$

Here are my proofs.

## formal languages ​​- Prove that \$ texttt {prefix} (L) \$ is normal

Given that $$L = lbrace 0 ^ n1 ^ n: n geq 0 rbrace$$ is a non-contextual non-regular language, proves that $$texttt {prefix} (L)$$ is regular.

Until now, I've planned that grammar to produce this language is:
$$S rightarrow 0S1 thinspace | thinspace epsilon$$

Would you like to prove $$texttt {prefix} (L)$$ is regular, as you would with any language, which proves that $$Sigma ^ star$$ = $$texttt {prefix} (L)$$, or by induction on the length of words in $$texttt {prefix} (L)$$.

## abstract algebra – Prove that \$ mathbb {Z}[sqrt{-2}]\$ is a Euclidean domain and \$ mathbb {Z}[sqrt{-10}]\$ n is not

Prove it $$mathbb {Z} [ sqrt {-2}]$$ is a Euclidean domain and $$mathbb {Z} [ sqrt {-10}]$$ is not.

I know that in general, to prove that something is a Euclidean domain, I have to prove the existence of a divisional algorithm involving a norm. In the case of $$mathbb {Z} [ sqrt {-2}],$$ the norm is $$a ^ 2 + 2b ^ 2.$$ I know how to prove that whole Gaussian numbers $$mathbb {Z} [ sqrt {-1}]$$ is a Euclidean domain, but I am not sure that the evidence on this subject relates to this evidence.

In addition, prove that $$mathbb {Z} [ sqrt {-10}]$$ This is not a Euclidean domain involves determining which ideals are not key, but I'm not sure how to find a non-main ideal. I think that should be generated by at least two elements of $$mathbb {Z} [ sqrt {-10}]$$ although.

As I am a novice in abstract algebra, I would like, if possible, more than just a hint.

## Prove inequality \$ sum _ { text {cyc}} frac {a} {a ^ 2 + b ^ 3 + c ^ 3} le frac1 {5abc} \$

Let $$a, b, c> 0$$ to be three real numbers such as $$a + b + c = 1$$. I want to prove that

$$frac {a} {a ^ 2 + b ^ 3 + c ^ 3} + frac {b} {b ^ 2 + a ^ 3 + c ^ 3} + frac {c} {c ^ 2 + a ^ 3 + b ^ 3} the frac {1} {5abc}.$$

My attempt: The use of AM-GM on each denominator gives (here, LHS refers to the left side)
$$LHS le frac {1} {3a ^ { frac12} bc} + frac {1} {3b ^ { frac12} ac} + frac {1} {3c ^ { frac12} ab}.$$

However, I think the last phrase may become larger than $$frac {1} {5abc}$$. Since if we multiply with $$3abc$$ then the initial inequality is:
$$sqrt a + sqrt b + sqrt the frac35$$

However, as $$a, b, c < 1$$, $$sqrt a + sqrt b + sqrt c> a + b + c = 1> frac35.$$

## lo.logic – The addition of an axiom of descriptibility on a theory similar to that of MK can it prove the existence of inaccessible sets?

Language: Logic of single-sorted first-order predicates

Primitives: $$=; in, emptyset, 0$$ where the last two are constants.

To define: $$set (x) equiv_ {df} exists y (x in y)$$

Axioms: ID +

1. Low extensionality:
$$non-empty (x) land forall z (z in x leftrightarrow z in y) to x = y$$

2. Foundation: $$exists y (y in x) to exists y in x ( no exists c (c in y earth c in x))$$

3. Class comprehension: if $$phi$$ is a formula in which $$x$$ is not free, so all closures of: $$exists x forall y (y in x leftrightarrow set (y) land phi)$$; are axioms.

To define: $$x = {y | phi } equiv_ {df} forall y (y in x leftrightarrow set (y) land phi)$$

4. Empty sets: $$0 neq emptyset land forall x (empty (x) leftrightarrow x = 0 lor x = emptyset)$$

5. Twinning: $$forall sets x, y (set ( {x, y }))$$

6. Sub-assemblies: $$set (x) land y subseteq x to set (y)$$

7. Power: $$set (x) to set ( mathcal P (x))$$

8. Union: $$set (x) to set ( bigcup (x))$$

To define: $$x = y ^ o equiv_ {df} x = o lor (not empty (x) land forall z in TC (x) (empty (z) to z = o))$$

Or $$TC$$ means transitive closure.

9. Modified size limit: $$| x ^ o | <| V ^ o | set (x ^ o)$$

Or $$V ^ o = {x ^ o | set (x ^ o) }$$

To define:

$$x ^ 0 cong y emptyset equiv_ {df} TC (x ^ 0) is in$$$$isomorphic to TC (y ^ emptyset)$$

Or $$TC$$ means transitive closure.

10. Describability: if $$phi ^ 0$$ is a parameterless formula in which all variables are set to 0, and if $$phi ^ emptyset$$ is the formula obtained from $$phi ^ 0$$ simply replacing each occurrence of the symbol 0 with the symbol $$emptyset$$then:

$${x ^ 0 | phi ^ 0 } cong {x ^ emptyset | phi ^ emptyset } to {x ^ 0 | phi ^ 0 }, {x ^ emptyset | phi ^ emptyset } text {are sets}$$

, is an axiom.

Question: would this theory prove the coherence of "ZFC + there is an inaccessible set"?

## Set theory – Prove that every bijection of the set in itself is representable in the form of a composition of two symmetries.

Task: prove that each bijection of the set in itself is representable as a composition of two symmetries.

Proof (where I am now and looking for ideas):

1) Write the task in mathematical language

Let $$f$$ be bijection of the whole $$X$$ in itself: $$f: X leftrightarrow X$$.

Let $$h$$ to be a symmetry on the board $$X$$ that means: $$h: X leftrightarrow X$$ and $$h ^ {- 1} = h$$.

Let $$g$$ to be a symmetry on the board $$X$$ that means: $$g: X leftrightarrow X$$ and $$g ^ {- 1} = g$$.

2) Need to prove that $$forall f: X leftrightarrow X exists (h: X leftrightarrow X, h ^ {- 1} = h ; and ; g: X leftrightarrow X, g ^ {- 1} = g): f = g circ h.$$

3) proof

Prove that there is $$h$$ and $$g$$ so that $$f = g circ h$$.

Here is where I get into the problem.
We know that:

1. the composition of two bijections is bijection.
2. $$(g circ h) ^ {- 1} = h ^ {- 1} circ g ^ {- 1}.$$
3. $$f ^ {- 1} = (g circ h) ^ {- 1} = h ^ {- 1} circ g ^ {- 1} = h circ g.$$
4. $$(h ^ {- 1}) ^ {- 1} = h.$$
5. the bijection of the set is a permutation of the elements of this set.

Help me find the right way to prove it for each bijection $$X leftrightarrow X$$ there is a way to find two compound symmetries do the same mapping as $$f$$.

## Prove that Rice's theorem does not apply to a property

This is related to a mission, but I would still appreciate some help to formalize the proof by private message or on this subject.

The question is whether Rice's theorem applies to certain properties. For example, for a structural property such as the number of states. I would say that:

• The property is decidable by its definition in coded form.
• The property is unrelated to the language, because with the same language we can easily find another MT that does not have this property, that is, with empty states. The property is therefore not a language property, and therefore Rice's theorem does not apply, because by definition, Rice's theory only applies if it is true. 39 is a non-trivial language property.

For me, this seems like a good argument, but the question is to award a substantial score of 10, and I do not think it's enough for 10 points. Am I missing out on a rigorous argument that I need to include to give my explanation? Clear as water of rock?