What are the factors affecting research?

Building research facilities will always increase the research output. If you build them on planets with research bonusses and on tiles with extra research bonusses they will be very effective. If build them on planets with research penalties and on normal tiles their output will be poor. Upgrading to more advanced research facilities helps a lot.

You need a prospering economy to research well. If you run into excessive deficit your research facilities will no longer be funded, which makes them stop output research.

I opened the Colonies Tab in then Civilization Manager tab, and I summed the total flasks of my planets. The number was 64, but when I opened my Finance Management Tab in my Domestic Stats, the research spending was 52. I opened the Research Tab, and the sum of research spending was the same as that stated in the Finance Management Tab (52). So, which one is correct? 65 or 52? Is the number of flasks not related to the number of research spending? And how does the research treaty affect the calculation of my research? could you explain the inner working of this problem? thanks a lot

I'm not sure if your Qs are broad or narrow.  For example, you did not mention Research bonuses in race set-up or from Research mining bases.  Also, if Research works in a parallel fashion with Morale and some others, some Research rate gains may happen when you research certain things even if you do not build them.  When one researches Banks, for example, there is an Economic gain empire-wide even if one does not build banks to get their bonus on the planets upon which they are built.  There're also some facilities which increase Research on a planet even though the structure(s) do not generate Research.  The Tech Capital is one example.

On a different note, are you sure you have the relationship down on sliders and Research?  IIRC, one must have the Research slider all the way to 100% to get the full output of any Research structure, just as one must have the slider to 100% to get the full output of a factory to build improvements, or to 100% to get the full output to build ships.  Have you moved the sliders around and gone to the Colonies tab to see the effects with various positions?  It is possible to have the same Research output from, say, 3 Research structures as 6, simply by having the slider at 100% for the 3 and 50% for the 6.  Of course, you are paying the same per building Maintenance on the 3 or the 6 no matter where the sliders are.  Although, with the 6 at 50%, you can have the other 50% output going to factories building structures or ships.

When you say "no difference in my research progress," are you talking about the RP you see in reports or the number of turns it takes you to finish a tech? If you are playing the latest update of TA, then you might have hit one of the research cost hikes that happen periodically as your total research grows. (Basically, research slowly gets more expensive.)

If the time-to-tech is not part of your question, are probably in the often-confusing Land of Sliders that LTjim describes. In addition to the 'partial production' thing caused by your divisions among Military, Social, and Research spending, there's also the total spending slider, which is not set to 100% by default (I think it starts around 60?).

What I meant about "no difference" was the number of weeks left to complete a research. I was researching Basic Aquatic World Colonization. Before I sold all of my research centres, the number was 22 weeks left. after I sold them all, the number remains the same (22). And the total flasks were 64 and after the sale, 20. This is confusing.

I have seen the following behavior in "time left" numbers indicated.

When moving sliders around, the "time left" updates as I adjust.  Some find that a useful way to apportion spending.

When changing the count of facilities, or upgrading them, the "time left" does not update until the turn button is toggled.  I suspect that the algorithm the game uses for that calculation does not update the count and type of facilities during the turn itself, though it may change the number in the equation when it goes to apply it.  That is, it updates for display the % of production for the facilities but not the count.

So, if I have guessed right, a slider setting of 75% research on 10 facilities each generating 10 points would display the effects of 750 points, or (0.75)(10 facilities)(10 points per facility).  But, if you demolished one facility, the equation for the display would not change to reflect that there are now 9 facilities - being limited to using the value that existed at the start of the player turn.  Nonetheless, when the game turn ends, the actual end of turn values would be used.

I see. so basically it's just some bug that the number wasn't updated? but in the inside, the research centres still adds to my research progress.

What's the corelation between tax rate and spending rate? when I tried to move about the tax rate slider, the spending amount goes up and down accordingly. I thought that the spending rate is just the percentage of population hired by the empire. if the tax < spending, then the deficit amount would simply be deducted straight from your treasury. thanks

Hi!

You seem to be new here. I suggest you to check wiki: https://www.galciv.wikia.com/wiki/Galactic_Civilizations_Wiki. Lots of useful explanations there.

BR,  Iztok