One-way function is not injective when it is in NP

Let us $ \Sigma = \{0,1\}$ and $ f: \Sigma^* \rightarrow \Sigma^* \in FP$ for which is valid that $ \exists k: \forall x \in \Sigma^* : \lvert x \rvert ^ {1/k} \leq \lvert f(x) \rvert \leq \lvert x \rvert ^ k$ . Thus, we can say that the function $ f$ is one-way function.

We have language $ L = \{ w \; | \; \exists z \in \Sigma^*, w = f(z)\}$ . The question is, how to prove that $ f$ is not injective if $ L \in NP \setminus UP$ , where $ UP$ is the class of unambiguous TM.

It is clear, that if $ L \in NP \setminus UP$ then exists NTM which decides this language and may exist at least one acceptable path for $ w \in L$ .

Minimal injective coresolution in the stable Auslander algebra

Let $ A$ be a finite dimensional (connected) quiver algebra. Let $ T(A)$ denote the full subcategory of coherent functors from $ mod-A$ to $ Ab$ that vanish on projective objects. $ T(A)$ is equivalent to the module category of the stable Auslander algebra of $ A$ when $ A$ is representation-finite.

The indecomposable projective objects of $ T(A)$ are $ \underline{Hom_A}(-,X)$ for any indecomposable non-projective object $ X$ and this is non-injective if and only if $ X$ has codominant dimension equal to zero. So assume in the following that $ X$ is non-projective with codominant dimension zero (meaning that the projective cover of $ X$ is not injective).

Auslander and Reiten showed (see for example their article “stable equivalence of artin algebras”) that a injective coresolution of $ \underline{Hom_A}(-,X)$ is given as follows when $ 0 \rightarrow \Omega^1(X) \rightarrow P \rightarrow X \rightarrow 0$ determines the projective cover of $ X$ :

$ 0 \rightarrow \underline{Hom_A}(-,X) \rightarrow Ext_A^1(-,\Omega^1(X)) \rightarrow Ext_A^1(-,P) \rightarrow Ext_A^1(-,X) \rightarrow Ext_A^2(-,\Omega^1(X)) \rightarrow Ext_A^2(-,P) \rightarrow Ext_A^2(-,X) \rightarrow Ext_A^3(-,\Omega^1(X)) ….$

Questions:

  1. In general this injective coresolution will not be minimal. Is it known what the minimal injective coresolution looks like?

  2. When is the above injective coreslution minimal (or at least minimal until the first term is zero)? I suppose that this is for example the case when $ P$ and $ \Omega^1(X)$ are indecomposable.

  3. Is there are formula for calculation the injective dimension of $ \underline{Hom_A}(-,X)$ or at least when it is finite?

  4. In case every indecomposable summand of $ X, P$ and $ \Omega^1(X)$ has infinite injective dimension, does also $ \underline{Hom_A}(-,X)$ has infinite injective dimension?

Check whether $f \mapsto f+ \frac{df}{dx}$ is injective or surjective!!

Consider maps $ C^{\infty} \to C^{\infty}$ s.t $ f \mapsto f+ \frac{df}{dx}$ . We have to check whether this map is injective or surjective.

My try: The map is clearly not injective as $ x$ and $ x+e^{-x}$ maps to $ x+1$ .

Now to check whether the map is subjective. Consider $ g \in C^{\infty}$ . Then I was thinking in this way that considering $ \int_0^xg$ then $ f=g-\int_0^xg$ now $ f+\frac{df}{dx}=g-\int_0^xg+\frac{dg}{dx}-g=-\int_0^xg+\frac{dg}{dx}$ still I am not getting a proof whether it is surjective or not!!

About maxima of injective holomorphic maps on $\mathbb{C}^n$

I am hoping the following is true. Mention of related ideas/topics are appreciated.

Suppose $ F:\mathbb{C}^n \to \mathbb{C}^n$ is a injective holomorphic mapping such that $ F(0)=0$ and $ dF(0) = I_n$ where $ I_n$ is the $ n \times n$ identity matrix. Let $ \partial B$ denote the boundary of the unit ball centered at the origin in $ \mathbb{C}^n$ . Let $ M = \sup_{x \in \partial B} ||F(x)||$ where $ || \cdot ||$ is the usual Euclidean norm. Then $ \partial B \cap \{x: M = ||F(x)|| \}$ is equal to one of three things: i) $ \{p\}$ for some point $ p$ , ii) $ \{\alpha p : |\alpha|=1 \}$ for some point $ p$ , iii) $ \partial B$

How can I prove that some functions are injective?

Definition of an injective function

$ $ f(x_1) = f(x_2) => x_1 = x_2 $ $

Well I have two functions that I can’t prove…

$ $ f(x) = x^3 + x $ $ $ $ f(x) = \frac{x}{1-log(x)} $ $ log(x) denotes the logarithm base 10 function, well it doesn’t matter for this problem, but whatever.

the first function $ $ x_1^3 + x_1 = x_2^3 + x_2 $ $ What can I do after this? Same with the second function, how can I manipulate in order to achieve $ $ x_1 = x_2 $ $

Injective map on spectrums

I know that if we have a surjection $ f:B\rightarrow A$ , this induces an injection on the spectrums $ f^* Spec A\hookrightarrow Spec B.$

What about the opposite? Does an injection in the spectrums of affine schemes induce a surjection in rings?

I would assume if such a nice property holded I would have found it somewhere online, so can you provide a counterexample? Also, is there a natural set of conditions such that if $ f^*$ satisfies them, then we can conclude that $ f$ is a surjection?

Injective resoution of the ring of entire functions

Let $ R$ be the ring of entire functions. I heard that the concrete value of the global dimension of the ring depends on continuum hypothesis. I would think that the injective dimension of the regular module $ R$ should coincide with the global dimension for this ring but Im not sure.

Question: Can one write down an explicit (minimal) injective resolution of the regular module $ R$ ? How does it look like in case it is possible?