DFT Vs DCT - Spectrum Differences

Have a look at Wikipedia. There is an example image, but for DFT

1. Has science fully discovered the spectrum of light?

Hmm, probably not! What is above the microwave frequency?

2. Consciousness - on/off switch or spectrum? [closed]

On/off is indeed not the commonest view of consciousness. In Hindu philosophical circles, consciousness is thought of as continuous across living systems. Humans have 'budding'consciousness, (plus free will), animals 'shuttered' and plants 'covered'consciousness, and these are points along a continuum. (Reference example: Hinduism and Science, by T D Singh; a paper in "Science and Religion: Global Perspectives", June 4-8, 2005, in Philadelphia, PA, USA, a program of the Metanexus Institute)

3. Behavior of the resolvent near the boundary of the spectrum

Let $H^2(D)$ be the Hardy space of holomorphic functions on the unit disk $D$. That is, $f in H^2(D)$ iff $f = sum_n=0^inftyf_nz^n$ has a power series expansion in $D$ with $|f|^2=sum_n=0^infty|f_n|^2 1$, regardless of $f in L^2(T)$, which supplies a counter-example to your conjecture.

4. Prove that the spectrum of normal matrices is stable under small perturbations

We know that $sigma(Aepsilon B)=sigma(A)mathcal O(epsilon)$. Write $B$ as sum over its components: $B=sum_ijB_ije_i e_j^*$.Then, $sigma(Aepsilon B)=sigmaleft(Asum_ijepsilon B_ij e_i e_j^*

ight)$, thus we can just apply $sigma(Aepsilon e_i e_j^*)=sigma(A)mathcal O(epsilon)$ $N^2$ times to obtain the result.

5. A first order sentence such that the finite Spectrum of that sentence is the prime numbers

This exists due to very general results, namely that the set of primes is rudimentary. See this excellent survey on spectra: Durand et al. Fifty Years of the Spectrum Problem: Survey and New Results. The same holds true for all known "natural" number theoretic functions. Indeed, the authors remark in section 4.2 that "we are not aware of a natural number-theoretic function that is provably not rudimentary"

6. Which signals are unsafe on the spectrum?

This question is vague. If you mean the entire electromagnetic spectrum, the only wavelengths that are inherently unsafe are the ones with ionizing radiation - X-rays and gamma rays.Any other wavelength is unsafe only if the power density is high enough to produce heating. And of course visible and IR wavelengths are unsafe for the eye if you stare at the sun. Which signals are unsafe on the spectrum?

7. How do spectrum gaps relate to topological protected states?

First "topological protected states" are states of many bodies.The gapped spectrum of excitations may have two interpretations: (A) The band structure has gaps. (In this case the "spectrum" is a single-particle spectrum) (B) The excitations on top of many-body ground state have a gap. (In this case the "spectrum" is a many-particle spectrum)For the case (A), the gapped spectrum (the gapped band structure) has nothing to do with "topological protected states", since the bands can be empty and there may not be any particles. Only the free fermion systems with filled bands may correspond to "topological protected states". The free fermion systems with filled bands is a special case of case (B). For the case (B) the gapped many-body ground states may correspond to "topological protected states". For details, see What does it mean for a Hamiltonian or system to be gapped or gapless?For the present question, one needs to clarify "topological protected state" of what particles? Is it the topological protected state of the atoms that form the material? In this case, the state is not even topological, since the state is not gapped.Is it the topological protected state of the phonons with a gapped band structure? In this case the state is not even topological, since there is no phonons at T=0.(There are many misuses of "topological" to describe things that are not topological.)

8. why does hydrogen atom exhibit a line spectrum instead of a continuous spectrum?

Photon emission and absorption is quantized! Those lines represent the energy it takes to move an electron from one orbital to another