derivatives – Why don’t introductory calculus textbooks often introduce the notion of differentiability on closed intervals?

The title says it all. I find apparent neglect of this idea rather unfortunate because (1) the notion of a function being differentiable on a closed interval is intuitively reasonable (and ought to be discussed), and (2) several theorems from calculus can be strengthened if they’re stated in terms of closed interval differentiability.

Before proceeding further, it’s worth taking time to make sure we agree on what we mean we say that a function is “differentiable over a closed interval”. Here’s the definition I had in mind:

A function $f:(a,b)tomathbb{R}$ is be differentiable over $(a,b)$ if it is differentiable (in the ordinary sense) over $(a,b)$, right-differentiable at $a$, and left-differentiable at $b$.

As you can probably guess, a function $f$ is said to be left-differentiable at some point $x_0$ if the limit

$$lim_{hto 0^-}frac{f(x_0+h)-f(x_0)}{h}$$

exists, with a similar definition applying to right-differentiability. The limits are then defined to be the left-hand and right-hand derivatives of $f$, respectively.

With that said, here’s an example of a theorem (FTC1) that, under its usual hypotheses, can be strengthened (albeit slightly) if the notion of closed-interval differentiability is used:

  • If $f:(a,b)tomathbb{R}$ is continuous, then the function $F:(a,b)tomathbb{R}$ defined by $F(x)=int_{a}^{x}f(t)text{ }dt$ is differentiable over $(a,b)$. Moreover, for all $xin(a,b)$, $F'(x)=f(x)$ (at the endpoints, $F'(x)$ is understood to be a left or right-hand derivative)

The return you get by applying this notion to this example is obviously minimal. That said, I truly believe that making these extra definitions is worthwhile anyway, namely because the resulting concepts strongly align with our intuitions.

To bring everything back together, I’ll restate my question one last time: why don’t introductory calculus textbooks often introduce the notion of differentiability on closed intervals? Any and all responses are greatly appreciated.

Relations between many category theory structures – introductory texts

There are many kinds of category theory structures. What are some good texts and good ways to remember the relations between them? For example, can there be a web of embedding relations between these category theory structures? (where can I find those webs to read.) Like which ones contain the other ones as more general cases. (Intersections and unions between different category theory structures.)

  • unitary braided fusion category = unitary premodular category

  • unitary fusion category

  • unitary symmetric fusion category

  • unitary modular category

  • monoidal category = tensor category

  • spherical fusion category

  • modular tensor category

Here are some facts I know of


text{unitary braided fusion category = unitary premodular category} supset
text{(1) unitary symmetric fusion category}\
text{(2) unitary modular category}

  1. The center construction of spherical fusion category defines a modular tensor category.

Please feel free to make comments and give a list of advices.

dg.differential geometry – What would be a good introductory reference for learning jet-bundle theory?

I am interested in learning the theory of Jet bundles, and am aware of the standard reference “The geometry of jet bundles” by D. J. Saunders. However this appears to be a detailed book, suitable for those who wish to specialise in this area. Can somebody recommend a relatively more introductory book (for a reader who knows the necessary differential geometric pre-requisites for learning this subject, but has never encountered Jet bundles) ? Thanks so much !

What introductory material can you recommend before buying a QNAP QuTS hero ZFS NAS?

Can anyone refer me to or recommend introductory reading material that would cover the following questions:

  • What would be a typical disk layout? From what I‘ve seen so far, it could be

    • 2 x SSD RAID-1 for the operating system
    • 2 x SSD RAID-1 for a read-write cache
    • The rest are HDDs, e.g. in a RAID-6 (RAID-Z2)

    How large would you choose the SSDs?
    Do the SSDs also get ZFS as file system?

  • At which level is disk encryption done? Are the system volume and the cache volume also encrypted?

  • How do I determine the memory requirement? There is a 32GB and a 128GB version of the TS-h1277XU-RP. Of course, the services on the virtualisation station need memory, but the unknown for me is the amount of memory needed for ZFS operations like deduplication, self healing, and snapshots.

  • What role does ECC RAM play? And is it only supported by the Intel version (TS-h1283XU-RP-E2236-128G)?

  • With the RAID-5 configuration in my previous NAS, RAID rebuilds after changing a drive always mean to hope that the remaining drives would survive the rebuild. So I would like to add another layer of redundancy with the new configuration: Would you go with RAID-6 (RAID-Z2), rely on cold-storage dumps, or do something else? How do I estimate the typical rebuild time of a ZFS RAID-5 (RAID-Z1) or RAID-6 (RAID-Z2) on these devices?

  • Is this a hardware RAID or a software RAID? How long do the RAID controllers last and can they just be swapped, keeping the existing data drives? Is the RAID controller monitored to detect possible problems? (I had this once in a server, where a RAID-controller problem showed up in the boot log and in a diagnostic tool and I ended up replacing the mainboard and the RAID controller while keeping the drives as they were.)

  • Is extending an existing RAID-6 (RAID-Z2) in size a problem? I was thinking of leaving some of the drive slots empty to add more drives over the next 5 years. But this QNAP intro video at 19:00 min sounds like one can’t just add another drive to an existing RAID. I think, RAID-Z expansion was planned to land in 2020. Is it there, yet? Can one trust it? What are the alternatives?

  • What flavor of ZFS are they using in QuTS? Do I need to buy a license to use ZFS?

  • How does the virtualization station perform? How does, for example, running postgres and redis databases on the virtualization station compare to running those within a docker environment on a separate server?

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    enter image description here

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    Introductory details to demonstrate union language closures

    Given a transition function $ delta colon Q times Sigma to Q $The standard argument is defined by defining a extended transition function $ has { delta} colon Q times Sigma ^ * to Q $ (after the notation in Sipser)
    this affects strings (see J.E. Hopcroft, R. Motwani, J.D. Ullman. Introduction to automaton theory, languages ​​and calculations 3ed., Section 2.2.4 for reference).
    This function is defined recursively
    $$ has { delta} (q, varepsilon) = q qquad hbox {and} qquad
    hat { delta} (q, Wa) = delta bigl ( hat { delta} (q, W), a bigr) $$
    to the $ W in Sigma ^ * $,
    $ a in Sigma $ and $ q in Q $and it's a simple exercise (with proof outlined below) to show this (as defined in Sipser)
    a finite automaton $ M = (Q, Sigma, Delta, q_0, F) $ accepts a string $ w in Sigma ^ * $ then and only if $ has { delta} (q_0, w) in F $.

    The heart of our desired inductive argument lies in the following result.

    To let $ w in Sigma ^ * $. Then, with notation as in the proof of Theorem 1.25 in Sipser,
    we have
    $$ hat { delta} bigl ((r_1, r_2), w bigr)
    = bigl ( hat { delta} _ {! 1} (r_1, w),
    has { delta} _ {! 2} (r_2, w) bigr). $$

    Proof. We go on by induction $ | w | $. The result is derived from the definitions of when $ | w | = $ 0
    as we have to have $ w = varepsilon $. Assume that the result is inductive for everyone $ w $ With $ | w | <n $.
    Then given $ w in Sigma ^ * $ With $ | w | = n $can we write $ w = w_1 dots w_n =: Ww_n $, in order to
    $$ begin {align}
    hat { delta} bigl ((r_1, r_2), w bigr)
    & = hat { delta} bigl ((r_1, r_2), Ww_n bigr) \
    & = delta Bigl ( hat { delta} bigl ((r_1, r_2), W bigr), w_n Bigr) \
    & = delta Bigl ( bigl ( hat { delta} _ {! 1} (r_1, W), hat { delta} _ {! 2} (r_2, W) bigr), w_n Bigr) \
    & = Bigl ( delta_1 bigl ( hat { delta} _ {! 1} (r_1, W), w_n bigr),
    delta_2 bigl ( hat { delta} _ {! 2} (r_2, W), w_n bigr) Bigr) \
    & = bigl ( hat { delta} _ {! 1} (r_1, Ww_n),
    has { delta} _ {! 2} (r_2, Ww_n) bigr) \
    & = bigl ( hat { delta} _ {! 1} (r_1, w),
    has { delta} _ {! 2} (r_2, w) bigr)
    end {align} $$

    as needed.

    With the above result, it is easy to complete the proof of correctness.
    Indeed since $ F = (F_1 times Q_2) cup (Q_1 times F_2) $, it follows that
    $$ has { delta} (q_0, w) in F $$
    then and only if
    $$ has { delta} _ {! 1} (q_1, w) in F_1 qquad hbox {or} qquad hat { delta} _ {! 2} (q_2, w) in F_2. $$
    In other words, $ M $ recognizes a string $ w in Sigma ^ * $ then and only if $ M_1 $ or $ M_2 $ recognize it.

    Addendum. We now outline a proof that the two definitions of a finite state machine that accepts a string are equivalent. The first definition can be found on page 40 of Sipser. We'll repeat it here for simplicity:

    To let $ M = (Q, Sigma, Delta, q_0, F) $ be a finite automaton and let $ w = w_1w_2 dots w_n in Sigma ^ * $ be a string. Then $ M $ accepted $ w $ if $ w = varepsilon $ and $ q_0 in F $, or when $ n> 0 $ and a sequence of states $ r_0 $,$ r_1 $, …, $ r_n $ in the $ Q $ exists so that $ r_0 = q_0 $, $ delta (r_i, w_ {i + 1}) = r_ {i + 1} $ to the $ 0 le i le n-1 $, and $ r_n in F $.

    So we have to show the following:

    A finite automaton $ M = (Q, Sigma, Delta, q_0, F) $ accepted $ w in Sigma ^ * $ then and only if $ has { delta} (q_0, w) in F $.

    Evidence sketch. If $ M $ accepted $ w = w_1 dots w_n $, then $ hat { delta} (q_0, w) = delta bigl ( hat { delta} (q_0, w_1 dots w_ {n-1}), w_n bigr) = delta (r_ {n- 1}, w_n) = r_n $ by an inductive argument; intuitively, this corresponds to showing that the recursive definition of $ has { delta} $ got & # 39; right & # 39; Are defined. Conversely if $ has delta (q_0, w) in F $Then we can create the sequence $ r_0 = q_0 $, $ ; r_1 = delta (r_0, w_1) $, … inductive.