[*]I wanted to know the possible affordable solutions to outside backup your shared hosting solution. I have some options like
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Payment Software Solutions
ag.algebraic geometry – Existence of Solutions to Quadratic Vector Equation
Given
$$A_{i,j,k} X_j^* X_k + C_i = 0$$
where $𝐴_{i,j,k}$ and $𝐶_i$ are arbitrary complex numbers for all $𝑗, 𝑘$ which are $𝑁$-dimensional indices and $i$ which is an $m$-dimensional index where $m<N$. Note $*$ is complex conjugation and there is a summation for repeated indices. Are there conditions on $A$ and $C$ to guarantee the existence of a solution to this set of equations?
ap.analysis of pdes – Pointwise estimate of solutions to the parabolic equation with a monotonic drift
I wonder for a parabolic equation
$u_t+(a(t,x)u)_x= u_{xx}$,
if we know that $a(t,x)$ is monotonic decreasing in $x$ with $a(t,-infty)=C_L, a(t,+infty)=C_R$, $C_L>C_Rgeq 0$, are there results developed to give precise pointwise estimates for $u$? Or long-time behaviors? I bet that $||u||_{L^{infty}}leq C/sqrt{t}$.
What are the Good Practices project parameters in SSIS (SQL) solutions
My question is not something concrete with an error in the X code. Rather, I would like to know if there are popularly known good practices when configuring the parameters in an SSIS project / solution. For example, if apart from parameterizing folder paths of inputs and outputs or connections, it is also common to put analysis dates or perhaps the dates fit better as a variable.
On the other hand, let’s say that in a certain environment I just found that the parameters of a set of SSIS packages instead of being configured as project parameters are in an SQL table and that is where they are modified. Is this normal or is it recommended to save as parameters?
Thanks
number theory – Positive integer solutions to $pxy+x+y=p^{#}$
Let $p$ be prime and $p^{#}$ the product of all primes not larger than $p$. Are there any positive integers $x$ and $y$ such that $pxy+x+y=p^{#}$. It appears there are no solutions. There are no solutions with $p<200$. Can it be shown that this is the case for all primes $p$?
My thoughts: Let $n=pxy+x+y$. If we can show that if $q | n$ for all primes $q le p$ and $qnmid n$ for all primes $q>p$ then $q^2 | n$ for some prime $q le p$, then we are done. (Note: The proposition is false for higher powers of $p^{#}$. It’s also false when $p^{#}$ is replaced by $p^{#} + b, b not = 0$)
differential equations – NDEigensystem solutions depend on how many solutions I ask for?
Background
I am using NDEigensystem to solve the following eigenvalue problem:
$$ left( begin{matrix} m&-ipartial_x \ -ipartial_x & -mend{matrix}right) left( begin{matrix} u_u(x) \ u_d(x) end{matrix} right)=lambda left( begin{matrix} u_u(x) \ u_d(x) end{matrix} right),qquad text{with} quad m = begin{cases} m = -10, , x leq 0 \ m = 10, , x>0end{cases} $$
I am specifically looking for solutions that vanish at $xrightarrow pm infty$.
Code Implementetion
My code is the following
mass1d(m1_, m2_, x_) := m2 UnitStep(x) - m1 UnitStep(-x) (* mass function *)
nSols = 10;(* number of solutions *)
{vals, sols} = NDEigensystem({
(* Diff Equation *)
-I ud'(x) + mass1d(10, 10, x)*uu(x),
-I uu'(x) -mass1d(10, 10, x)*ud(x),
(* Boundary conditions*)
DirichletCondition(uu(x) == 0, True),
DirichletCondition(ud(x) == 0, True)
}, {uu(x), ud(x)}, {x, -3, 3}, nSols)(* SOLVER *) ;
Problem/Question
It is my understanding that NDEigensystem will return the solutions with the lowest (absolute value) eigenvalues. Hence, if I run my code asking for 5 solutions and then asking for 10 solutions, I would expect the 5 solutions in the first run also appear in the second run. This is not what happens. What I obtain is the following:
Aslo, when I ask for only 5 solutions I obtain the following warning that is not present when I ask for
Eigensystem::chnpdef: Warning: there is a possibility that the second matrix SparseArray(Automatic,<<2>>,{1,{{0,4,9,<<74>>,300,302},<<1>>},{0.08 +0. I,-0.01+0. I,0.02 +0. I,0.02 +0. I,-0.01+0. I,0.08 +0. I,<<290>>,0.16 +0. I,0.02 +0. I,0.02 +0. I,0.16 +0. I,0.02 +0. I,0.16 +0. I}}) in the first argument is not positive definite, which is necessary for the Arnoldi method to give accurate results.
Furhtermore, this problem has an analytical solution, which is $lambda = 0$ and functions $u_u(x) sim e^{-x}$ and $u_d(x) sim e^{-x}$, which tells me asking for 10 solutions gives the correct answer.
Why do I get this warning only when I ask for 5 solution, but not when I look for 10?
Complete code
Above I write the way am finding the solutions, below is the complete code just to show there is no difference in how am running both cases:
*(* ----- 5 SOLUTIONS ------ *)
nSols = 5;(* number of solutions *)
{vals5, sols5} = NDEigensystem({
(* Diff Equation *)
-I ud'(x) + mass1d(10, 10, x)*uu(x),
-I uu'(x) - mass1d(10, 10, x)*ud(x),
(* Boundary conditions*)
DirichletCondition(uu(x) == 0, True),
DirichletCondition(ud(x) == 0, True)
}, {uu(x), ud(x)}, {x, -3, 3}, nSols)(* SOLVER *) ;
nSols = 10;(* number of solutions *)
(* ----- 10 SOLUTIONS -----*)
{vals10, sols10} = NDEigensystem({
(* Diff Equation *)
-I ud'(x) + mass1d(10, 10, x)*uu(x),
-I uu'(x) - mass1d(10, 10, x)*ud(x),
(* Boundary conditions*)
DirichletCondition(uu(x) == 0, True),
DirichletCondition(ud(x) == 0, True)
}, {uu(x), ud(x)}, {x, -3, 3}, nSols)(* SOLVER *) ;
(* PLOTING *)
GraphicsGrid(
{{
BarChart((ReIm /@ vals5)/Max({m1, m2}),
ChartLegends -> {"Real Part", "Imaginary Part"},
ChartLabels -> {Range(nSols), None},
PlotLabel -> "Asking for 5 solutions"),
Plot(Abs(sols5((1)))^2 /. {x -> x0} // Evaluate, {x0, -l, l},
PlotRange -> All,
PlotLabel -> "Asking for 5 solutions (1st result)",
PlotLegends -> {"!(*SubscriptBox((u), (u)))",
"!(*SubscriptBox((u), (d)))"})
},
{
BarChart((ReIm /@ vals10)/Max({m1, m2}),
ChartLegends -> {"Real Part", "Imaginary Part"},
ChartLabels -> {Range(nSols), None},
PlotLabel -> "Asking for 10 solutions"),
Plot(Abs(sols10((1)))^2 /. {x -> x0} // Evaluate, {x0, -l, l},
PlotRange -> All,
PlotLabel -> "Asking for 10 solutions (1st result)",
PlotLegends -> {"!(*SubscriptBox((u), (u)))",
"!(*SubscriptBox((u), (d)))"})
}
}
, ImageSize -> Large)*
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nt.number theory – On solutions to linear system amalgam
Input:
I. System of $Omega(t)$ linear polynomials in $mathbb F_2(x_1,dots,x_{t})$.
II. System of $Omega(t)$ linear polynomials in $mathbb Z(x_1,dots,x_{t})$.
-
Can we output a common $0/1$ solution of the system in polynomial time?
-
Can we output parity of the number of common $0/1$ solutions of the system in polynomial time?
Essentially we are input a polyhedron which is a subset of the Boolean cube and a system of linear equations in $mathbb F_2$.
matrix – Finding the first few solutions of an eigen equation that has infinitely-many solutions
I have a system that is described by a complex matrix, $U$, which is a function of two parameters $k, beta$ (a dispersion eigen-relation in wave physics). We are interested in the solutions ($k,beta$) that make the determinant of this matrix equal to zero, $det(U)=0$. So, for a given value of $k$, we try to search for the values of $beta$ that will make $det(U)=0$.
However, there is an infinite number of solutions ($beta$ values) for any given $k$, and when I try to Solve this in Mathematica, it keeps running forever, as expected. So I was wondering whether there is a way to ask Solve (or any similar function) to only give us the first $n$ solutions to the equation, where $n$ is some number chosen by us? Or to ask it to give us the nearest $n$ solutions in the neighborhood of an initial value for $beta$, say $beta=beta_{0}$?
