## parameterized complexity – Finding a kernel for d-Bounded degree deletion

In $$d$$ Bounded degree deletion problem, we are given an undirected graph $$G$$ and a positive integer $$k$$, and the task is to find at most $$k$$ such vertices whose removal decreases the the maximum vertex degree of the graph to at most $$d$$.

The question is to how to find a polynomial kernel (in $$k$$ and $$d$$) for this problem.

I seem to be able to get the only reduction rule that if any vertex has degree $$> k+d$$, it has to be there in the deletion set (if the answer to instance is yes). Because if it isn’t, then at least $$k+1$$ of its neighbors have to be in deletion set. I can’t seem to move beyond this point.

The exercise is from this book (exercise $$2.9$$).

I am also aware that we can remove edges between vertices with degree $$< d$$, and find solution in the modified graph (hint from the book). But I am not sure how it will be useful, in getting a bound over number of vertices/edges in $$k$$ and $$d$$.

I would appreciate only hints if possible (something maybe beyond the book hints).

PS: for $$d=0$$ this reduces to vertex cover problem.

## fa.functional analysis – Condition on kernel convolution operator

I am studying a about O’Neil’s convolution inequality. It is stated that for $$Phi_1$$ and $$Phi_2$$ be $$N$$-functions, with $$Phi_i(2t)approx Phi_i(t), quad i=1,2$$ with $$tgg 1$$ and $$k in M_+(R^n)$$ is the kernel of a convolution operator.

The $$rho$$ is an r.i. norm on $$M_+(R^n)$$ given in terms of the r.i norm $$bar rho$$ on $$M_+(R_+)$$ by
$$rho(f)=bar rho(f^*), quad f in M_+(R_+)$$

Denote Orlicz gauge norms, $$rho_{Phi}$$, for which
$$(bar rho_{Phi})_dapprox bar rho_{Phi}left(int_0^t h/tright).$$

It is stated that
$$rho_{Phi_1}(k+f)leq C rho_{Phi_2}(f)$$
if
$$(i) quad bar rho_{Phi_1}left(frac 1t int_0^t k^*(s)int_0^sf^*right)leq C bar rho_{Phi_2}(f^*)$$
$$(ii) quad bar rho_{Phi_1}left (frac 1tint_0^t f^*(s)int_0^sk^*right)leq C bar rho_{Phi_2}(f^*)$$
$$(iii) quad bar rho_{Phi_1}left(int_t^{infty}k^*f^*right)leq C bar rho_{Phi_2}(f^*).$$

I cannot understand under which conditions on kernel those inequalities (i),(ii) and (iii) would hold.

dmitry@chicago:~$uname -a Linux chicago 5.4.0-48-generic #52-Ubuntu SMP Thu Sep 10 10:58:49 UTC 2020 x86_64 x86_64 x86_64 GNU/Linux Hello everyone! My HP laptop has 3 USB ports. After hibernation a brand new mouse doesn’t respond. When I change port from 3 to 2 or 1 the mouse works perfect. When I attach USB-Flash drive in port 3 it also works. Only mouse doesn’t want to work on port 3 after hibernation. Which commands outputs should I present to give you more info? ## How can I tweak my kernel to use all CPUs in my rooted device? I have Huawei ALE-L21 alice (2015) Magisk-rooted Bootloader-Unlocked Xposed-Installed but it’s somehow slower and I decided to tweak my CPU but here is what ES Kernel Manager shows: Also, I can’t actually use the governor This is my ROM info: ## crash – Suggestion bar still causing Kernel crashes? On Mathematica 12.1.1, Windows 10. When Suggestions bar is active, entering and evaluating the following crashes the Kernel. zz = {{{0, 2}, {1, 1}, 28, 3}, {{1, 1}, {3, 1}, 56, 1}, {{1, 2}, {3, 1, 1}, 168, 0}}  Seeking confirmation before tagging as bug and reporting to WRI. ## How can Monolithic kernel based OS are faster the Microkernels? I have been studying about OS and currently, I am on "types of the kernel". Now in the book and some websites are saying Monolithic kernel-based OS is faster, but how are they faster than Microkernel-based OS. ## Kernel panic 0x00007f00 – Ask Ubuntu I tried to boot my ubuntu machine and this error message appears. I read through dozens of previous questions about this and no one answered how to boot to the system. Please, I need to boot to the system. I have work to deliver tomorrow. I will be fired if I cant get access to my computer. It is the only one I have. ## performance tuning – Parrallel Computation – how to send only necessary data to kernel? Let’s try to do simple test: 1. Make one-kernel run. data = Table(Pi i /50000., {i, 1, 100000}); ndata = AbsoluteTiming@Table(Total(data((1 ;; i)))^(2/3), {i, 1,100000});  2. Launch two sub-kernels and move all data to each of them LaunchKernels(2) DistributeDefinitions(); ParallelEvaluate(MemoryInUse())  as you can see, they needs {52577800, 52578016} bytes of memory. 1. Run parallel calculations on two kernels ndata2 = AbsoluteTiming@ParallelTable(Total(data((1 ;; i)))^(2/3), {i, 1, 100000}); ParallelEvaluate(MemoryInUse())  It takes {53379616, 53379832} with newly calculated data. 1. Launch two more kernels,distribute all data again and check the memory: LaunchKernels(2) DistributeDefinitions(); ParallelEvaluate(MemoryInUse())  They take {53379616, 53379832, 53378328, 53378400} bytes and data distribution is fast enough relatively to calculation time. 1. Run again the calculations on four kernels: ndata4 = AbsoluteTiming@ParallelTable(Total(data((1 ;; i)))^(2/3), {i, 1, 100000}); ParallelEvaluate(MemoryInUse())  The memory consumption is still small: {53379616, 53379832, 53379696, 53379768} Finally, let’s check the timings of our three attempts: {ndata((1)), ndata2((1)), ndata4((1))} (* {6.9033127, 2.572234, 1.6979114} *)  As you can see, such simple type of parallelization requires proportionally more memory but it gives an acceleration for ~2.7 and ~4.06 times with two and four sub-kernels correspondingly. These numbers are bigger than 2 and 4 due to the internal vectorization of certain operations. When two cores of my CPU are formally free, you can see evident boost of the calculations (increase for 2.7 instead of 2). I especially did the inter-kernel data transfer separately to see the difference in pure performance. For really big data arrays and simple chain of operations per element it can spend more time for memory allocation than for calculations ## testing – Kernel test crashes with a DrupalCoreDependencyInjectionContainerNotInitializedException I have a kernel test that’s crashing with a DrupalCoreDependencyInjectionContainerNotInitializedException. The test involves a custom entity type which has a computed field. The computed field’s class uses createItem() to create a field value which holds computed data: class MyComputedFieldClass extends FieldItemList { use ComputedItemListTrait; /** * {@inheritdoc} */ protected function computeValue() {$this->list(0) = $this->createItem(0, ( 'value' => 'foo', )); } }  The backtrace shows that this is where the problem happens: 1) DrupalTestsmy_moduleKernelEntitlementsTest::testEntitlementTypeFixed PHPUnitFrameworkException: Fatal error: Uncaught DrupalCoreDependencyInjectionContainerNotInitializedException: Drupal::$container is not initialized yet. Drupal::setContainer() must be called with a real container. in /var/www/docroot/core/lib/Drupal.php:131
Stack trace:
#0 /var/www/docroot/core/lib/Drupal.php(159): Drupal::getContainer()
#1 /var/www/docroot/core/lib/Drupal/Core/Field/FieldItemList.php(41): Drupal::service('plugin.manager....')
#2 /var/www/docroot/modules/custom/my_module/src/Field/MyComputedFieldClass.php(30): DrupalCoreFieldFieldItemList->createItem(0, Array)
`

What’s going on?

## If the Mathematic is running the code in one kernel, how can I run another code in other kernel?

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