Let V be the set of positive reals, V = {x ∈ R : x > 0}. Define “addition” on V by x“ + ”y = xy, and for α ∈ R, define “scalar multiplication” on V by “αx” = x^α . Is V a vector space with these unusual operations of addition and scalar multiplication? Prove your answer.

Suppose T ∈ L(V)

a) Show that V = ImT⁰ ⊃ ImT¹ ⊃ ImT² ⊃ ...

b) Show that if m ≥ 0 is such that ImT^m = ImT^{m + 1}, then ImT^{m + k} =ImT^{m + k + 1} for all k ≥ 0.

c) Show that if n = dim V, then ImTⁿ = ImT^{n + 1} = ImT^{n + 2} = ....

Let V and W be Banach spaces and suppose T : V → W is a linear map. Suppose that for every f ∈ W∗ the corresponding linear map f ◦ T on V is in V ∗ . Prove that T is bounded.

The potential of a particle is given as V (x) = -Vδ (x), with V being a positive number. Find the wave function and energy of the bounded states of this particle.

Let v and w be two nonzero vectors in R4 . Then v and w are linearly independent if only if v is not a scalar multiple of w. True or false?

The potential in a region between x = 0 and x = 6.00 m is V = a + bx, where a = 12.6 V and b = -7.90 V/m.

(a) Determine the potential at x = 0.
______ V

Determine the potential at x = 3.00 m.
______ V

Determine the potential at x = 6.00 m.
______ V

(b) Determine the magnitude and direction of the electric field at x = 0.

Determine the magnitude and direction of the electric field at x = 3.00 m.

Determine the magnitude and direction of the electric field at x = 6.00 m.

Problem 3. Throughout this problem, we fix a matrix A ∈ Fn,n with the property that A = A∗. (If F = R, then A is called symmetric. If F = C, then A is called Hermitian.) For u, v ∈ Fn,1, define [u, v] = v∗ Au. (a) Let Show that K is a subspace of Fn,1. K:={u∈Fn,1 :[u,v]=0forallv∈Fn,1}. (b) Suppose X is a subspace of Fn,1 with the property that [v,v] > 0 for all nonzero v ∈ X. (1) Show that [−, −] defines an inner product on X. (c) Suppose Y is a subspace of Fn,1 with the property that [v,v] < 0 for all nonzero v ∈ Y. (2) If X is a subspace with property (1), prove that X + K + Y is a direct sum, where K is defined in part (a).

The final product of the Calvin Cycle (the product that EXITS the Calvin Cycle, assuming successful carboxylation) is

To be in an active state, several enzymes of the Calvin Benson cycle require

The dry-weight mass of a plant (the weight of plant solids after water is removed) is from

Which statement is correct

The antennae complexes of photosynthesis gather light energy and channel it toward (choose ALL that apply)

Assuming linear electron flow from H₂O to NADP⁺
(i.e., the Z-scheme) what is the theoretical minimum number of photons required to produce one molecule of O₂ evolved (remember each water release a pair of electrons, it takes 2 H₂O to produce O₂)?

During CYCLIC electron flow, electrons from ferrodoxin are transferred to

What the heck am I talking about when I say “Reduction of plastoquinone to plastoquinol occurs on the stroma side of the thylakoid membrane, and oxidation of plastoquinol back to plastoquinone occurs on the lumen side of the thylakoid membrane”? This describes:

The Calvin-Benson cycle takes place predominantly in the

Put the following in the correct order for linear photosynthesis (the Z-scheme)

CF-type ATPase

Plastoquinone/plastiquinol

Photosystem II

Photosystem I

Cytochrome b6f

Plastocyanin

   I put economics, because this question is from my econometrics review, I can change it to statistics if needed

Question 1 Consider the following sample of 6 observations measuring students’ test scores (score) and average hours spent on studying each week (study).
   Table 1

Suppose that you are interested in estimating the following model:
   score = β0 + β1study + u.

(i) Use the method of OLS to estimate β0 and β1. Show your work.

(ii) What are the predicted test scores if a student spends 8 hours and 20 hours per week studying, respectively? Comment on your predictions

(iii) Find the sum of squared residuals, standard error of residuals, and the standard error of ˆ β1.

(iv) Calculate and interpret R^2.

(v) Suppose that you include in (1) another regressor, sleep, which is average hours spent sleeping per week. All the 6 students in your survey reported that they sleep 8 hours. Can you apply OLS to estimate the effect of sleep time on test scores? Explain.

San Jose Mercury News, March 31, 2020: “As classes go online, students want Stanford to lower tuition”: “Thousands of Stanford students are asking for a discount in tuition for this quarter --- up to 80% --- as, (as one student leader stated): “the online education will affect our EDUCATIONAL EXPERIENCE – (the caps are mine) – interacting with faculty and peers … (is)… a more ESSENTIAL PART of our time spent at Stanford” …. Online education “makes us UNABLE TO ENJOY AN EDUCATIONAL EXPERIENCE QUALIFIED FOR THE $10,000 (tuition only) that we spent for this Spring quarter”…. “if the experience is not IMMERSIVE, then IT IS NOT FULL AND DOES NOT DESERVE THE PRICE TAG”……. QUESTIONS!

A. Should students be given a discount on tuition this quarter (or semester) owing to classes going online? (you did not SIGN UP for this, right???) Why or why not? At Stanford? At EVC? Is there a difference?

B. Is your answer any different for GRADUATE students in law and medicine? Why or why not? C. Stanford has offered to let students take a quarter off, thus delaying… THE REST OF THEIR LIVES…..in your opinion, why may this solution be… ‘less than optimal’? D. Is it relevant that Stanford is sitting on MANY TENS OF BILLIONS OF DOLLARS IN ENDOWMENTS, --- one of the highest in the nation -- and could EASILY AFFORD to give their students a discount of 80% for one quarter?

E. WHAT SHOULD STANFORD DO? Offer an 80% discount? 100%? 50 %? 30%? No discount at all? Should it vary depending on the INCOME of the student and her family---- (just like the tuition itself, and room and board expenses)?

F. May we agree that students who were living in the dorms should be reimbursed for their HOUSING costs, given that they were KICKED OUT (evicted)? What about students who had been renting apartments from landlords in and around Palo Alto? Should they be treated differently? Why or why not? (Many landlords will suffer a huge drop in income in 2020) G. WHAT HAS BEEN YOUR MOST CHALLENGING OBSTACLE, IN TERMS OF TAKING AND COMPLETING YOUR CLASSES ONLINE, SINCE MARCH 10, 2020? IN YOUR OPINION, HOW HAS THE QUALITY OF YOUR EDUCATION CHANGED?

H. IS THIS ISSUE IN YOUR ‘TOP TEN’ LIST OF PROBLEMS RIGHT NOW, IN YOUR OPINION? Top 5? Top 3? WHY OR WHY NOT?