How to make a polynomial division of arbitrary non-negative power in Mathematica?

Suppose I have a simple polynomial in $ {a, b } $, defined as $ a ^ k-b ^ k forall k in mathbb {Z} ^ { geq0} $. If I know that one of the factors is $ (a-b) $is there a way to get a representation of its remaining factors in Wolfram Language?

I know that one of his representations is $ sum_ {j = 0} ^ {k-1} {(a ^ {(k-1) -j} b ^ j)} $ and Mathematica recognizes that

Sum[a^((k-1)-j) b^j,{j,0,k-1}]


(a ^ k - b ^ k) / (a ​​- b)

But if I ask


He is unable to do anything.

Also, what is the right way to formulate hypotheses?

Is the expression above interpreted differently if I give as

Supposing[k[Element]NonNegativeIntegers, FullSimplify[(a^k-b^k)/(a-b)]]

Also tried to factoring directly without giving a single factor without success,

Supposing[k[Element]NonNegativeIntegers, Factor[a^k-b^k]]

$ x $ is a non-negative integer and $ sqrt {x ^ 2 + sqrt {x + 1}} $ is a positive integer.

Find a non-negative integer $ x $ such as $ sqrt {x ^ 2 + sqrt {x + 1}} $ is a positive integer

Because $ sqrt {x ^ 2 + sqrt {x + 1}}> x $, we leave $ x ^ 2 + sqrt {x + 1} = (x + y) ^ 2, (y> 0) $

That means $$ begin {aligned}
& x ^ 2 + sqrt {x + 1} = x ^ 2 + y ^ 2 + 2xy \
& implies sqrt {x + 1} = y ^ 2 + 2xy \
& implies x + 1 = y ^ 4 + 4x ^ 2y ^ 2 + 4xy ^ 3
end {aligned} $$

And that 's where I was stuck.

Max vs. bound min for non-negative harmonic function

Problem: Let $ Omega $ to be an open, bounded, simply connected subset of $ mathbb {C} $ and let $ u colon Omega to mathbb {R} $ to be a non-negative harmonic function. Show that for each compact subset $ K subseteq Omega $ there is a constant $ C_K> $ 0 it depends on K $ such as
$$ sup_ {x in K} u (x) leq C_K inf_ {x in K} u (x). $$

It seems to me that if $ u $ There are no zeros in $ Omega $, so we just take $ C_K = sup_ {x in K} u (x) / inf_ {x in K} u (x) $. But if $ u $ has a zero in $ Omega $then $ u $ reaches its minimum in $ Omega $ since $ u geq 0 $, So $ u = $ 0 by the minimum principle. So it seems we do not have $ Omega $ to be open and connected.

linear algebra – non-negative irreducible matrices with random (correlated or independent) nonzero entries

allows $ M $ to be a non-negative irreducible matrix. According to Perron-Frobenius' theorem, the maximum eigenvalue of $ M $, $ lambda $, is positive and equal to its spectral radius $ rho (M) $.

Suppose now the matrix $ M $ it is not deterministic and its non-zero elements are equal to the random variables $ tanh (x_i) $ with $ x_i sim N (m> 0, sigma ^ 2) $. However, the null elements are the same deterministic zeros as before. My question is: what will happen to the expected value of the maximum eigenvalue if $ x_i $The s are correlated in case they are independent.

My observation is that the existence of a positive correlation between non-zero inputs increases the expected maximum eigenvalue compared to the case where the inputs are independent. But I can not justify this experience. – The index was out of reach. It must have a non-negative value and less than the size of the collection. vb

Public Sub Form1_Load (sender as object and as EventArgs) handles MyBase.Load

Me.DataGridView1.Columns.Add ("A", "Account")
Me.DataGridView1.Columns.Add ("B", "Account Name")
Me.DataGridView1.Columns.Add ("C", "Must")
Me.DataGridView1.Columns.Add ("D", "Haber")
Me.DataGridView1.RowCount = 5

End Sub

I stated in the data grid 4 columns and 5 rows, but when performing this procedure, an error occurred.

Public Sub fillDatagrdid (
Me.DataGridView1 (0, 0) .Value = 12345
End Sub

It's as if the datagird had no cells, help me please!

Dg.differential geometry – Automatic Plurisubharmonicity for a non-negative function

I am confused on one point in this very short document. At the top of page 3, it is stated that:

Yes $ S $ is a totally real subvariety in an almost complex compact variety $ (X, J) $, then any function $ rho ge $ 0 near $ S $ (not degenerate transversely), disappearing on $ S $, must be strictly $ J-Purisubharmonic.

We say one two form $ theta $ strictly $ J-plurisubharmonic if for any $ v neq $ 0 we have $ theta (v, Jv)> $ 0.

The explanation of the author is very basic and I am confused. It seems like it's kind of a standard result, but I can not find any reference. I think MO experts should know that very well.

Thank you!

How to find as many as possible given a table of non-negative numbers in Java

Statement of the problem:
I have an array of non-negative string numbers in Java, I want to organize integers to form as many as possible.

Example: Below the entrance:

Chain[] numbers = {"15", "9", "62", "34"};

The arrangement "9623415" gives the greatest number.

Note: I understand that we can sort all the numbers in descending order, but sorting just does not work. For example, 15 is greater than 9 in the natural order, but "9" precedes "15" in the solution. What is the best way to implement a custom comparator in this case?

Any help would be appreciated.

dg.differential geometry – Distort the metrics of a non-negative Ricci curvature to a positive curvature

Given a closed Riemannian variety $ (M, g) $ with a non-negative Ricci curvature and $ dim geq $ 3when can we deform the metric into a positive Ricci curved curve?

I know it's impossible in general because of the flat factor in universal coverage. But what about we add some topological restrictions on $ M $ like just connectivity? Are there positive or negative results on this problem?

(In addition, are there now examples of closed varieties simply connected with a positive scalar curve metric to not admit a positive Ricci curve metric?)