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- Thread starter GAD
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"Holes occur when the numerator and denominator of a rational function have a common factor that cancels out during simplification." I hate to say it, but AI can be quite useful. Holes: also known as "a math thing."

I guess it's a common calculus word problem involving the rate of change of the volume of water in the tank, which decreases with the flow rate out of the hole.

Thanks, gang.

Yes this. The pressure at the bottom correlates with the flow out of the hole. As the water level drops due to the water flowing out the hole, the pressure drops, causing the flow of water to slow, leading to the rate of change of water pressure changing.I guess it's a common calculus word problem involving the rate of change of the volume of water in the tank, which decreases with the flow rate out of the hole.

I don't remember holes in bases, when I took calculus.

I definitely had that exact word problem. I think it was actually the first thing we did in Differential Equations.I don't remember holes in bases, when I took calculus.

I don't rememberI don't remember holes in bases, when I took calculus.

A course onabstract algebratypically covers fundamental concepts related to algebraic structures. The key topics often include:

Groups:

Definition and Examples: Understanding what a group is and exploring examples like integers under addition, non-zero real numbers under multiplication, and symmetry groups.Subgroups: Criteria for a subset to be a subgroup.Cyclic Groups: Groups generated by a single element.Group Homomorphisms: Structure-preserving maps between groups.Normal Subgroups and Quotient Groups: Understanding how to form new groups from existing ones.Group Actions: How groups act on sets, which can help in studying symmetries.Sylow Theorems: Results concerning the number of subgroups of a given order.Classification of Finite Groups: Basics of how finite groups are classified.Rings:

Definition and Examples: Introduction to rings, such as integers, polynomial rings, and matrix rings.Ring Homomorphisms: Maps between rings that preserve structure.Ideals and Quotient Rings: Analogous to subgroups and quotient groups in the context of rings.Integral Domains and Fields: Special kinds of rings where the multiplication operation has no divisors of zero and where every non-zero element has a multiplicative inverse.Factorization: Understanding prime elements, irreducibles, and unique factorization in rings.Fields:

Definition and Examples: Basic properties of fields like the field of rational numbers, real numbers, complex numbers, and finite fields.Field Extensions: Building larger fields from smaller ones and understanding their structure.Galois Theory (Advanced): Linking field theory with group theory to solve classical problems like solvability of polynomials.Modules (sometimes covered):

Generalization of Vector Spaces: Modules over rings, which extend the concept of vector spaces.Module Homomorphisms: Maps between modules that preserve structure.Exact Sequences: Short and long exact sequences in module theory, which play a role similar to chain complexes in topology.Additional Topics:

Algebraic Geometry (Introductory): Relation between algebra and geometry through the study of polynomial equations.Representation Theory (Advanced): Studying groups by representing their elements as matrices and analyzing the resulting structures.

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Witches stairs!

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And which is not?Witches stairs!

Nope. Fun for cats stairs.

Or more boringly, steps to be taken one leg per step ONLY.

Or more boringly, steps to be taken one leg per step ONLY.

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Pretty long stairway. Going up was not the problem for me, it was the coming down...

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And as GAD says, they're WITCH'S STAIRS!!

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You left out Skoda!

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That is some impressive tile work.