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# Non constructible functions and anomalous results

In the Arora-Barak book, in the definition of time constructible functions, it is said that using functions that are not time constructible can lead to "anomalous results". Does anyone have an example of such an "anomalous result"? I have heard in particular that there might exist functions such that the time hierarchy theorem does not hold, does anyone have an example of such functions? Is there something about this somewhere in the litterature?

Borodin's Gap Theorem: For every total computable function $g(n) \geq n$, there is a total computable function $t$ such that $DTIME[g(t(n))] = DTIME[t(n)]$.

In fact, this holds for any Blum complexity measure in place of $DTIME$.

Since the Wikipedia article doesn't give the proof and the paper is on ACM DL I thought that it might be useful to post the proof here:

THEOREM 3.7. (Gap Theorem).

Let $\Phi$ be a complexity measure, $g$ a nondecreasing recursive function such that $\forall x, g(x) \geq x$. Then there exists an increasing recursive function $t$ such that functions computable of complexity measure $t$ are the same as functions computable of complexity measure $g\circ t$.

PROOF.

Define $t$ as follows:

$$t(0) := 1$$ $$t(n) := \mu{ k > t(n-1)}: \forall i \leq n, (\Phi_i(n)<k \lor \Phi_i(n)> g(k) )$$

1. for all $n$, there is a $k$, since for all $i \leq n$:

a. if $\Phi_i(n)$ is undefined then $\forall k, \Phi_i(n)>g(k)$, and

b. if $\Phi_i(n)$ defined then $\exists k, \Phi_i(n) < k$.

2. $k$ can be found recursively, since $\Phi$ is a complexity measure and thus $\Phi_i(n) <k$ and $\Phi_i(n) > g(k)$ are recursive predicates.

3. $t$ satisfies the theorem, since $n \geq i$ implies that either $\Phi_i(n) < t(n)$ or $\Phi_i(n) > g \circ t(n)$.

QED.

We observe that an arbitrarily large $t$ can be found to satisfy Theorem 3.7. Suppose we want $t(n) > r(n)$, then define

$$t(0) := r(0) + 1$$ $$t(n) := \mu{k > max\{t(n-1), r(n)\}}: \ldots$$

(from <NAME>, "Computational complexity and the existence of complexity gaps", JACM 1972, with slight modifications.)

• The idea is to define $t(n)$ to be the least $k$ s.t. any function (of index less than $n$) that is computable in complexity measure $g(k)$ is also computable in complexity measure $k$. Oct 13 '11 at 7:47


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IF A=\begin{bmatrix} -1 & 0 & 2 \\ 3 & 1 & 2 \end{bmatrix} and B=\ | Filo

Class 12

Math

Algebra

Matrices

536
150

IF and then

1. and both exist
2. exists but bot
3. exists but not
4. but and do not exist
Solution: Since, order of is and order of is
Thus, and exists.
536
150

Connecting you to a tutor in 60 seconds.

format: plain


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# Higher Ramanujan equations II: periods of abelian varieties and transcendence questions

<NAME>

## Last Updated:

2019-04-01T18:17:37.3+01:00

## abstract:

In the first part of this work, we have considered a moduli space $B_g$
classifying principally polarized abelian varieties of dimension $g$ endowed
with a symplectic-Hodge basis, and we have constructed the higher Ramanujan
vector fields $(v_{kl})_{1\le k\le l \le g}$ on it. In this second part, we
study these objects from a complex analytic viewpoint. We construct a
holomorphic map $\varphi_g : \mathbf{H}_g \to B_g(\mathbf{C})$, where
$\mathbf{H}_g$ denotes the Siegel upper half-space of genus $g$, satisfying the
system of differential equations $\frac{1}{2\pi i}\frac{\partial \varphi_g}{\partial \tau_{kl}}=v_{kl}\circ \varphi_g$, $1\le k\le l \le g$.
When $g=1$, we prove that $\varphi_1$ may be identified with the triple of
Eisenstein series $(E_2,E_4,E_6)$, so that the previous differential equations
coincide with Ramanujan's classical relations concerning Eisenstein series. We
discuss the relation between the values of $\varphi_g$ and the fields of
periods of abelian varieties, and we explain how this relates to Grothendieck's
periods conjecture. Finally, we prove that every leaf of the holomorphic
foliation on $B_g(\mathbf{C})$ induced by the vector fields $v_{kl}$ is
Zariski-dense in $B_{g,\mathbf{C}}$. This last result implies a "functional
version" of Grothendieck's periods conjecture for abelian varieties.

968732


text:
An element x exists as two isoptopes -144x and 146x and weigh 143.95 amu and 145.95 amu, respectively. if the

Question:

an element x exists as two isoptopes -144x and 146x and weigh 143.95 amu and 145.95 amu, respectively. if the atomic weight if x us 145.23 amu, determine the percent abundabce of heavier isotope

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# Which of these methods do you employ?

This topic is 4957 days old which is more than the 365 day threshold we allow for new replies. Please post a new topic.

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Hi, im kinda writing my own general library of classes for basic 3d/2d shapes, like quads, panels etc, For more simplicity when managing these files in VC++ i wanna put the entire class in 1 CPP file, like the class def and body. What i wanna know is, is there any performance drop in having to include the whole class rather than just the class def, ie the .H file. ace

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To my knowledge the only performance issue caused by that is longer compile times.

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yeah thats what i imagined, thanx, can anyone verify this?

ace

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That's true, and you'd get a handful linker errors unless you put the code inside the class definition.

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well what i want to do is put the code outside the class definition but inside the same cpp file, can this be done?

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Yes, this can be done, of course. However, you will probably run into the following errors:

- If you try to use the class somewhere else than in the cpp file it is written in, you will get a compiler error.

- This compiler error can be solved by including the cpp file with the class definition inside the file that needs it. This will cause a linker error, because the member function definitions will appear in two cpp files.

- This linker error can only be solved by not including the member function definitions in a cpp file. This, again, is usually perfomed by 1° writing the member definitions inside the class definition or 2° putting member definitions in a file that does not get included anywhere. You don't want to do any of those, so you're stuck.

This can be summarized as: if you do put everything in a single cpp file, it will compile and link correctly, but you won't be able to use the contents of that file anywhere else.

EDIT: well, actually you can, using the following trick:

In each source file do the following:

#ifdef INCLUDING//all included files for this file here//here, your class definition#else#define INCLUDING//all included files for this file here, again#undef INCLUDING//here, your class definition, again//here, the definitions for the members of your class#endif

In addition to include guards and the like. This works, but takes more time than traditional .cpp/.h implementation and interface separation.

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If you want to put it all in 1 file then why insist on it being the .cpp file? It is infinitely better to do that in the .h file. All class methods go inside the definition, and use include guards (as you should).

btw do you mean 'performance drop'? In the compiler, or in your built exe?
If you put it all in the .h file as I just described then there should be no compile time performance hit and it should work nicely.

If you really want it in your cpp file then you will probably have to make the class a template class. However you shouldn't ever be including .cpp files anywhere!

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yeah well i hadnt considered pttuing it all in .h files, but ive just gone with the separates now, thanx for all ur help guys

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iMalc: what is the real difference between .h and .cpp files, except for their name? ;)


text:
The Effects of Aquarobic Exercise Program on Body Composition and Blood Lipid Concentrations in Obese Elderly Females

Title & Authors
The Effects of Aquarobic Exercise Program on Body Composition and Blood Lipid Concentrations in Obese Elderly Females
<NAME>; <NAME>;

Abstract
In the elderly, a lower muscle mass and higher body fat mass are induced by a lower level of physical activity. A negatively changed body composition with an advanced age can lead to a higher falling risk and rate of diseases. On the other hand, aerobic-type exercise positively influences the body composition and hyperlipidemia in the elderly. Therefore, this study examined the effects of aquarobic training for 12 weeks on the body composition and blood lipid levels in obese old women. The subjects (n = 31, body fat: 33.42%) completed a 12 week water based aerobic training at 12 - 13 of Borg Scale of intensity (three times per week, each session: an hour). The body composition was measured by bioelectrical impedance analysis (Inbody 770-Biospace, Seoul, Korea) and the concentrations of blood lipids (high-density lipids cholesterol (HDL), low-density lipids cholesterol (LDL), triglyceride (TG) total cholesterol (TC)) were determined at pre and post training. A paired t-test was used for data analysis with $\small{{\alpha}=0.05}$. In the results, the body composition (% body fat (p < 0.05) and waist circumference (p < 0.05)) were reduced significantly. The LDL (p < 0.05), TG (p < 0.001) and TC (p < 0.05) were reduced significantly. In conclusion, a 12 week aquarobic exercise program helps improve the body composition and concentrations of serum lipids. Therefore, aquarobic exercise can enhance lipolysis using fat as energy to induce an improvement of the body composition and induce hyperlipidemia.
Keywords
aquarobic exercise program;body composition;blood lipid;elderly women;abdominal fat;
Language
Korean
Cited by
References
1.
<NAME>, <NAME>, The Effect of 20-Weeks' Aquarobics Exercise on the Body Composition, Physical Fitness, Blood Lipid and Glucose in Obese Elderly Women, The Korean Journal of Sports Science vol. 23, no. 1, pp. 1263-1272, 2014.

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<NAME>, <NAME>, Effectiveness of Exercise Training for a Short Period on Functional Fitness and Depression in the Frail Elderly, Journal of The Korean Society of Living Environmental System vol. 11, no. 2, pp. 122-128, 2004.

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<NAME>, Effects of Aquarobics for 48weeks on Body Composition, Health Related Fitness and Metabolic Syndrome Risk Factors in Elderly Obese Woman, The Korean Society of Sports Science, vol. 23, no. 4, pp. 1191-1200, 2014.

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<NAME>, <NAME>, <NAME>, <NAME>, <NAME>, and <NAME>, The Study on the Characteristics of Obese Elderly Based on Blood Test, Journal of Korean Medicine for Obesity Research, vol. 13, no. 2, pp. 66-73, 2013.

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<NAME>, <NAME>, <NAME>, and <NAME>, The effect of obstacle training in water on static balance of chronic stroke patients, Journal of Physical Therapy Science, vol. 2, no. 6, pp. 437-440, 2014. DOI: http://dx.doi.org/10.1589/jpts.26.437

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<NAME>, <NAME>, <NAME>, <NAME>, <NAME>, <NAME>, <NAME>, <NAME>, and <NAME>, Effects of aquatic exercise training using waterresistance equipment in elderly, European Journal of Applied Physiology, vol. 108, no. 5, pp. 957-964, 2010. DOI: http://dx.doi.org/10.1007/s00421-009-1306-0

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<NAME>, <NAME> and <NAME>, A specific inpatient aquatic physiotherapy program improves strength after total hip or knee replacement surgery: a randomized controlled trial, Archives of Physical Medicine and Rehabilitation, vol. 90, no. 5, pp. 745-755, 2009. DOI: http://dx.doi.org/10.1016/j.apmr.2008.12.011

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<NAME>, <NAME>, and <NAME>, The Effect of Aquatic Gait Training on Foot Kinesiology and Gait Speed in Right Hemiplegic Patients, The Journal of the Korea Contents Association, vol. 9, no. 12, pp. 674-682, 2009. DOI: http://dx.doi.org/10.5392/JKCA.2009.9.12.674

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<NAME>, <NAME>, <NAME>, <NAME>, and <NAME>, Effects of aquatic exercise on flexibility, strength and aerobic fitness in adults with osteoarthritis of the hip or knee, Journal of Advanced Nursing, vol. 57, no. 2, pp. 141-152, 2007. DOI: http://dx.doi.org/10.1111/j.1365-2648.2006.04102.x

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<NAME>, The Effects of Aquarobics Exercise on Body Composition and Isokinetic Muscular Function of Postmenopausal Patients with Knee Osteoarthritis, Journal of exercise rehabilitation, vol. 7, no. 1, pp. 133-142, 2011.

20.
<NAME>, The effects of aquarobics exercise programs for female elderly suffering from degenerative arthritis on pain, bone mineral density and leg muscle strength, The Korea Joumal of Sports Science, vol. 21, no. 1, pp. 803-814, 2012.

21.
<NAME>, The Effects of Self-Consciousness on the Body Image and Self-Esteem for Middle-Aged Women Participating in Aquarobics, The Korea Journal of Sports Science, vol. 20, no. 3, pp. 221-234, 2011.

22.
<NAME> and <NAME>, The Effects of an Aquarobics Exercise Program on Body composition and Inflammation Markers in Obese Middle-aged Women, The Korean Alliance for Health,Physical Education, vol. 52, no. 3, pp. 483-493, 2013.

23.
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24.
<NAME>, <NAME>, <NAME>, <NAME> and <NAME>, Effects of Aquarobics Exercise on Body Composition, Fitness and Health Related Quality of Life(SF-36)in Elderly Women, Korean Gerontological Society, vol. 30, no. 3, pp. 683-694, 2010.

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<NAME>, <NAME> and <NAME>, The Effect of Aquarobic Exercise on the Apo E, Leptin and Blood Lipids of Obese Women, Korea Sports Research, vol. 18, no. 1, pp. 299-308, 2007.

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<NAME>., <NAME>., <NAME>. & <NAME>. (2002). Lipids, lipoproteins, and exercise. J Cardiopulm Rehabil, vol. 22, no. 6, pp. 385-398. DOI: http://dx.doi.org/10.1097/00008483-200211000-00002

28.
<NAME>, The Effect of an Aquarobic Exercise Program on the Self-Efficacy, Pain, Body Composition, Blood Lipid and Depression in women with Osteoarthritis, Chonbuk National University, 2009.

29.
<NAME>, <NAME>, <NAME> and <NAME>, Effects of Silverobics Exercise on Metabolic Syndrome Risk Factors, Bone Mineral Density and Living Fitness of Elderly Women, The Korea Joumal of Sports Science, vol. 21, no.1, pp. 1155-1164, 2012.

30.
<NAME>, Effect of aquarobic exercise on body composition and blood lipids of obese women, Chonnam University, 2003.

31.
<NAME>, Effect of Aquarobic Exercise on Metabolic Syndrome and Physical Fitness of Elderly Women, Chonnam University, 2010.

32.
ACSM.(2006). ACSM'S Guideliness for Exercise Testing and Prescription. 6th Edition. Philadelphia: Lippincott, Williams & Witkins.

33.
<NAME>, Effect of 12-week aquatic exercise on weight, percent body fat, serum glucose and serum lipids on elderly women, Kookmin University, 2008.

34.
<NAME>, The Difference of Physical Fitness and Blood Lipid by types of the Aquatic Exercise Programs in the Elderly Women, Korea National Sport University, 2007.

35.
<NAME>, <NAME> and <NAME>, The Effect of Aquarobics on Metabolic Syndrome Risk Factors in Middle-aged Women, Journal of Korean Physical Education Association for Girls and Women, vol. 23, no. 4, pp. 45-56, 2009.

36.
<NAME>, The effects of aquatic exercise on body composition and serum lipids in obese middle-aged women, Korean journal of physical eduaction, vol. 42, no. 2, pp. 519-527, 2001.

37.
<NAME>, <NAME>, <NAME>, <NAME>, <NAME> and <NAME>, The effects of 48 weeks aqua-exercise on blood lipid profile and body composition of elderly women, Korean journal of physical eduaction, vol. 40, no. 2, pp. 717-737, 2001.

38.
<NAME>, Effect of 16-week Aquarobics Exercise Program on Body Composition, Physical Fitness, and Blood Lipids of Aged Women, Korea National Sport University, 2007.


text:
# Solve $dy/dx = e^{x^2}$

I want to solve $$\frac{dy}{dx}=e^{x^{2}}.$$ i using variable separable method to solve this but after some stage i stuck with the integration of $\int e^{x^{2}}dx$. i dont know what is the integration of $\int e^{x^{2}} dx$. Please help me out!

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There is no closed form integral of $e^{x^2}$. Are you sure you don't have a typo? –  Arkamis Jul 20 '12 at 4:03
@Ed: it has a closed form. It's not elementary, however. –  J. M. is back. Jul 20 '12 at 4:10
Why in this green earth was this question downvoted? –  J. M. is back. Jul 20 '12 at 5:39
@J.M. Maybe someone was on the blue part of Earth. –  <NAME> Jul 20 '12 at 7:04
@Rahul: I wrote a long-ish answer to that question many moons ago, but the gist is, I consider the error function as a "known quantity", and I thus treat it as a closed form. –  J. M. is back. Jul 20 '12 at 7:43

$$\frac{dy}{dx}=e^{x^{2}}$$ has no elementary solution. The error function (also called the Gauss error function) is a special function (non-elementary) of sigmoid shape which occurs in probability, statistics and partial differential equations. It is defined as: $$\operatorname{erf}(x) = \frac{2}{\sqrt{\pi}}\int_{0}^x e^{-t^2} dt.$$ See the link for reference and more information and thus, J.M. ...?
...and thus, the solution of the DE is $y=-\frac{i\sqrt\pi}{2}\mathrm{erf}(ix)+C$. (the imaginary error function) –  J. M. is back. Jul 20 '12 at 5:41


text:
# Could we change the color of the accepted answer check for color blind people?

So I just figured out that stack exchange exists like 2 months ago and I am now just getting active in the wonderful world of cross validated and its great.

One thing I came across just now is I spent 10 minutes trying to figure out why I could not accept an answer. I was thinking I did not have the privilege, but I looked at the list and it is not on there (why would it be, you should be able to accept answers if you can ask questions). Then I sorted out that I had accepted an answer, but the color change from the grayed out check mark to the accepted check mark does not tickle the cones in my eye enough for me to notice the difference (rather, one set of cones in my eye is on permanent vacation and the other cones can't be bothered to take over the work, slackers, I know).

I realize that I can just hover over it and it will say accepted x amount of time ago, so its not the end of the world, but red/green color blindness is pretty common (other color blind folks would likely have an issue as well), so I was wondering if it would be possible to make it a different color (a darker more pronounced green would do it for me, though I can't speak for all color blind people). For an example of something that works, check out the darker green in stack overflow, that works just fine for me, though it might not fit the layout of cross validated as well.

I guess a color blind mode would solve the problem as well, but that seems like a bit of overkill, given that the only time it has bugged me so far is on the accept answer check.

• Thank you for bringing this to our attention. (I usually check for possible red/green confusion, but it did not occur to me that green could be confused with gray!) Perhaps @Jin could take a look? Probably a solution could be as simple as using the same (dark) green for the checkmark as for the voting symbols. – whuber Feb 29 '12 at 18:56
• Thanks for the response. Its strange when it comes up. The only times it matters for me is the red green color blind test (you know the circles), stop signs obscured by leaves (whoops!) and occasionally on websites. Hard to quantify exactly when on websites and it generally doesn't matter, but the fix is usually painless. I am terrible at color matching games though. – asjohnson Feb 29 '12 at 19:21
• I totally agree with asjohnson's as I have the same problem. Please do something about it. Darker green and or clearly lighter grey would be great (I think the best would be to change both colours). For me the basic issue is also that there is too little contrast between the colors. – Henrik Mar 4 '12 at 14:21
• Could this be fixed by, e.g. use of a larger tick mark when the answer has been accepted, or adding in a title "Accepted Answer" above the relevant answer? It would appear that a change in the form used to identify the accepted answer might be more useful. – Michelle Mar 26 '12 at 3:02
• I imagine a larger tick mark would be more noticable and text would be another solution that would get it done. I guess I would have to check the specific solution to be certain, but I don't see why either of them would not work. – asjohnson Mar 26 '12 at 13:51
• +1 It's a reasonable request--designers ought to pay attention to this issue. In the meantime, note that when you hover the mouse over an arrow corresponding to a vote you already cast, the hovertext will end in "(click again to undo)". – whuber Sep 4 '15 at 14:56
• I am confused by what you mean by saying that I have already had my votes locked on a few questions by trying to up-vote them when they were already up-voted. Trying to upvote an already upvoted or downvoted question does not lead to any consequences! You get a notification saying that your vote is locked. The vote is locked automatically after a short delay (I think it's 5 min) following the original voting; it is not the attempt to re-vote that locks it! – amoeba Sep 4 '15 at 15:13
• @amoeba thanks. That makes sense. I was confused by the message I was getting. Anyway I think I up-voted and then forgot and tried again because it looked similar to the neutral color. – pbible Sep 5 '15 at 1:50
• I'm not color blind but I have one crummy monitor that I can't tune really well so gradients are always washed out. On that monitor the colors are very close, and it is worse on Meta. – Erik Sep 18 '15 at 17:20
• @whuber I'd like to add that I too have trouble distinguishing between a pressed and unpressed vote arrow, as the color contrast ratio (CR) between the two is extremely poor. On the main site, the colors are Unpressed #818185 : #027187 Pressed, CR = 1.46:1, and on Meta, Unpressed #818185 : 3b3b3b Pressed, CR = 2.89:1. Both are failing ratios according to WCAG, which requires 4.5:1. Using [Flux](justgetflux.com) at night makes the CR even worse, due to the faint blue tinge of the buttons getting completely erased. – Iwillnotexist Idonotexist Sep 29 '15 at 21:39
• @AnnaLear, could you explain what has changed, after the tag status-completed was added into this question? We were talking about this in the chat (ten fold room), but we couldn't figure it out. Thanks. – <NAME> Oct 7 '15 at 12:49
• @AndreSilva ... I was on the wrong post. Sorry about that. – <NAME> Oct 7 '15 at 13:55

Maybe, there is a workaround for this.

I am using Stylish on Chrome to visualize accepted-answers on CV exactly equal to Stack Overflow. I think the green color used on the main site is too light.

The bellow code is a summary of what I got from blah238's answer on GIS SE, and <NAME>'s answer on Stack Overflow SE. And there is also the Old CheckMark's answer on Meta Stack Overflow SE.

Here it goes:

1- Install the Stylish on Google-Chrome (it can be used with Mozilla Firefox too, but not sure about other web navigators).

2- Type on Stylish the following code:

div.status.answered-accepted, div.status.answered-accepted strong {
background: rgb(117, 132, 92) !important;
color: #e1e818 !important;
}

background-color: rgb(117, 132, 92) !important;
color: #e1e818
}

.vote .vote-accepted-on {
background-position: 0% 0%;
}

The last part of the code refers to altering the checkmark. The checkmark image provided in the code is cutted a little bit but Brock Adams's answer provide an alternative way to one upload an own picture that can be used with background-image argument (the image needs to be uploaded on the server already with the desired color/aspect).

3- Still on Stylish set the "Applies to" field to:

URL starting with --> https://stats.stackexchange.com/

Here are some examples/screenshots for the above code: 1, 2 and 3.


text:
How to call covers not covering anything else?

Let $A$ be a set. Then it seems common to name cover of $A$ any set $\mathcal C$ of sets with $\bigcup{\mathcal C}\supseteq A$.

1. Is there then a good name for the particular covers $\mathcal C$ with $\bigcup{\mathcal C}=A$ ?

2. Are there standard names for those covers $\mathcal C$ with $\emptyset\notin\mathcal C$ ?

Question 2 also arises for partitions. Well-regarded books allowing the empty set to be a member of a partition (which I then would only call a quasipartition) but using the term differently later on include Kelley's and Dugundji's topology texts.

-

The construction "covering of $A$ by [something]" seems handy. For (1), one can use "covering of $A$ by subsets of $A$", and for (2) one can use "covering of $A$ by nonempty sets".
An answer to the first question is: An exhaustion of $A$.