## 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}]
``````

given

``````(a ^ k - b ^ k) / (a ​​- b)
``````

``````FullSimplify[(a^k-b^k)/(a-b),Assumptions->k[Element]NonNegativeIntegers]
``````

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.

## vb.net – 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.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
``````

## 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 $$dim geq 3$$when 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?)