## linear algebra – Trace inequality under consideration of definiteness

Let $$G in mathbb{R}^{3×3}$$ a symmetric, but indefinite matrix and $$U in mathbb{R}^{3×3}$$ a symmetric and positive definite matrix. Now, the following inequality shall be proved:
$$text{Tr}(G^2) leq text{Tr}(GUGU^{-1})$$
If $$U$$ and $$G$$ have the same eigenvectors, both sides of the inequality are obviously equal. However for more general cases,
I have tried to rearrange the inequality to
$$text{Tr}(underbrace{(UG-GU)}_{text{skew-symmetric}} GU^{-1}) leq 0$$
and then using the Cauchy-Schwarz inequality in order to achieve an upper boundary that possibly fulfils the inequality. Unfortunately, I have not found a solution yet.

## He is scamming me. Is there any way to trace his location?

Well I just found out that he has scammed a lot of people already. i was lucky enough to not be one of them. Im still interacting with him but he does not know that I know what he is up to. He pretends to be an employer and wants me to buy supplies from him through bitcoin as payment. I have the address where I would aend the bitcoin to. Is there any way to trace it back?

## magento2 – I faced the “Infinite loop detected, review the trace for the looping path” while place order in checkout page?

1 exception(s):
Exception #0 (LogicException): Infinite loop detected, review the trace for the looping path

Exception #0 (LogicException): Infinite loop detected, review the trace for the looping path

```#1 MagentoCheckoutModelSessionInterceptor->getQuote() called at (vendor/magento/module-checkout/Model/Cart.php:221)
#2 MagentoCheckoutModelCart->getQuote() called at (generated/code/Magento/Checkout/Model/Cart/Interceptor.php:76)
#3 MagentoCheckoutModelCartInterceptor->getQuote() called at (app/code/StripeIntegration/Payments/Helper/Generic.php:439)
#4 StripeIntegrationPaymentsHelperGeneric->hasSubscriptions() called at (app/code/StripeIntegration/Payments/Plugin/Tax/Config.php:22)
#5 StripeIntegrationPaymentsPluginTaxConfig->aroundGetAlgorithm() called at (vendor/magento/framework/Interception/Interceptor.php:135)
#6 MagentoTaxModelConfigInterceptor->MagentoFrameworkInterception{closure}() called at (vendor/magento/framework/Interception/Interceptor.php:153)
#7 MagentoTaxModelConfigInterceptor->___callPlugins() called at (generated/code/Magento/Tax/Model/Config/Interceptor.php:130)
#8 MagentoTaxModelConfigInterceptor->getAlgorithm() called at (vendor/magento/module-tax/Model/TaxCalculation.php:163)
#9 MagentoTaxModelTaxCalculation->calculateTax() called at (vendor/magento/framework/Interception/Interceptor.php:58)
#10 MagentoTaxModelTaxCalculationInterceptor->___callParent() called at (vendor/magento/framework/Interception/Interceptor.php:138)
#11 MagentoTaxModelTaxCalculationInterceptor->MagentoFrameworkInterception{closure}()
#12 call_user_func_array() called at (vendor/vertex/module-tax/Model/Plugin/TaxCalculationPlugin.php:69)
#13 VertexTaxModelPluginTaxCalculationPlugin->aroundCalculateTax() called at (vendor/magento/framework/Interception/Interceptor.php:135)
#14 MagentoTaxModelTaxCalculationInterceptor->MagentoFrameworkInterception{closure}() called at (vendor/magento/framework/Interception/Interceptor.php:153)
#15 MagentoTaxModelTaxCalculationInterceptor->___callPlugins() called at (generated/code/Magento/Tax/Model/TaxCalculation/Interceptor.php:26)
#16 MagentoTaxModelTaxCalculationInterceptor->calculateTax() called at (vendor/magento/module-tax/Model/Sales/Total/Quote/Subtotal.php:43)
#17 MagentoTaxModelSalesTotalQuoteSubtotal->collect() called at (vendor/magento/framework/Interception/Interceptor.php:58)
#18 MagentoTaxModelSalesTotalQuoteSubtotalInterceptor->___callParent() called at (vendor/magento/framework/Interception/Interceptor.php:138)
#19 MagentoTaxModelSalesTotalQuoteSubtotalInterceptor->MagentoFrameworkInterception{closure}()
#20 call_user_func_array() called at (vendor/vertex/module-tax/Model/Plugin/SubtotalPlugin.php:64)
#21 VertexTaxModelPluginSubtotalPlugin->aroundCollect() called at (vendor/magento/framework/Interception/Interceptor.php:135)
#22 MagentoTaxModelSalesTotalQuoteSubtotalInterceptor->MagentoFrameworkInterception{closure}() called at (vendor/magento/framework/Interception/Interceptor.php:153)
#23 MagentoTaxModelSalesTotalQuoteSubtotalInterceptor->___callPlugins() called at (generated/code/Magento/Tax/Model/Sales/Total/Quote/Subtotal/Interceptor.php:26)
#24 MagentoTaxModelSalesTotalQuoteSubtotalInterceptor->collect() called at (vendor/magento/module-quote/Model/Quote/TotalsCollector.php:274)
#26 MagentoQuoteModelQuoteTotalsCollector->collect() called at (vendor/magento/module-quote/Model/Quote.php:1995)
#27 MagentoQuoteModelQuote->collectTotals() called at (generated/code/Magento/Quote/Model/Quote/Interceptor.php:1064)
#28 MagentoQuoteModelQuoteInterceptor->collectTotals() called at (vendor/magento/module-checkout/Model/Session.php:269)
#29 MagentoCheckoutModelSession->getQuote() called at (generated/code/Magento/Checkout/Model/Session/Interceptor.php:63)
#30 MagentoCheckoutModelSessionInterceptor->getQuote() called at (vendor/magento/module-checkout/Controller/Onepage.php:153)
#31 MagentoCheckoutControllerOnepage->dispatch() called at (vendor/magento/framework/Interception/Interceptor.php:58)
#32 MageplazaOscControllerIndexIndexInterceptor->___callParent() called at (vendor/magento/framework/Interception/Interceptor.php:138)
#33 MageplazaOscControllerIndexIndexInterceptor->MagentoFrameworkInterception{closure}() called at (vendor/magento/framework/Interception/Interceptor.php:153)
#34 MageplazaOscControllerIndexIndexInterceptor->___callPlugins() called at (generated/code/Mageplaza/Osc/Controller/Index/Index/Interceptor.php:91)
#35 MageplazaOscControllerIndexIndexInterceptor->dispatch() called at (vendor/magento/framework/App/FrontController.php:159)
#36 MagentoFrameworkAppFrontController->processRequest() called at (vendor/magento/framework/App/FrontController.php:99)
#37 MagentoFrameworkAppFrontController->dispatch() called at (vendor/magento/framework/Interception/Interceptor.php:58)
#38 MagentoFrameworkAppFrontControllerInterceptor->___callParent() called at (vendor/magento/framework/Interception/Interceptor.php:138)
#39 MagentoFrameworkAppFrontControllerInterceptor->MagentoFrameworkInterception{closure}() called at (vendor/magento/module-store/App/FrontController/Plugin/RequestPreprocessor.php:99)
#40 MagentoStoreAppFrontControllerPluginRequestPreprocessor->aroundDispatch() called at (vendor/magento/framework/Interception/Interceptor.php:135)
#41 MagentoFrameworkAppFrontControllerInterceptor->MagentoFrameworkInterception{closure}() called at (vendor/magento/module-page-cache/Model/App/FrontController/BuiltinPlugin.php:73)
#42 MagentoPageCacheModelAppFrontControllerBuiltinPlugin->aroundDispatch() called at (vendor/magento/framework/Interception/Interceptor.php:135)
#43 MagentoFrameworkAppFrontControllerInterceptor->MagentoFrameworkInterception{closure}() called at (vendor/magento/framework/Interception/Interceptor.php:153)
#44 MagentoFrameworkAppFrontControllerInterceptor->___callPlugins() called at (generated/code/Magento/Framework/App/FrontController/Interceptor.php:26)
#45 MagentoFrameworkAppFrontControllerInterceptor->dispatch() called at (vendor/magento/framework/App/Http.php:116)
#46 MagentoFrameworkAppHttp->launch() called at (generated/code/Magento/Framework/App/Http/Interceptor.php:24)
#47 MagentoFrameworkAppHttpInterceptor->launch() called at (vendor/magento/framework/App/Bootstrap.php:261)
#48 MagentoFrameworkAppBootstrap->run() called at (index.php:39)
```

## fa.functional analysis – How are the vector-valued trace and the unique linearization of \$mathfrak L(X,Y):hatotimes_π:X→Y\$ of \$mathfrak L(X,Y)×X→Y,;(L,x)↦Lx\$ related?

Let $$X$$ be a $$mathbb R$$-Banach space and $$X’:hatotimes_pi:X$$ denote the completion of the tensor product of $$X’$$ and $$X$$ with respect to the projective norm. The trace functional $$operatorname{tr}:X’:hatotimes_pi:Xtomathbb R$$ is the unique linearization of the bounded bilinear form $$X’times Xtomathbb R;,;;;(varphi,x)mapstovarphi(x)tag1.$$ Now, I’ve seen that there are two related notions:

1. If $$E$$, $$F$$ and $$G$$ are $$mathbb R$$-vector spaces, $$C_G:(Eotimes G’)otimes(Gotimes F)to Eotimes F;,;;;(xotimes u’)otimes(uotimes y)mapstolangle u’,urangle_{G’,:G}xotimes ytag2$$ is called tensor contraction or vector-valued trace
2. If $$H$$ and $$K$$ are separable $$mathbb R$$-Hilbert spaces and $$T$$ is a trace-class operator on the Hilbert-Schmidt tensor product $$H:hatotimes_2:K$$, then there is a unique trace-class operator $$operatorname{tr}_K(T)$$ on $$H$$, called partial trace, with $$operatorname{tr}left(operatorname{tr}_K(T)Bright)=operatorname{tr}left(Tleft(Botimes_2operatorname{id}_Hright)right)tag3$$ for all $$Binmathfrak L(H)$$

(I guess that in 2. there is a similar construction possible for the projective tensor product; but I haven’t found any reference for that.)

Now, if I’m not missing anything, there should be a third thing we can construct: If $$Y$$ is another $$mathbb R$$-Banach space, we could define $$operatorname{tr}_{X,:Y}:mathfrak L(X,Y):hatotimes_pi:Xto Y$$ as the unique linearization of the bounded bilinear operator $$mathfrak L(X,Y)times Xto Y;,;;;(L,x)mapsto Lxtag4.$$ How are these notions (especially the tensor contraction and $$operatorname{tr}_{X,:Y}$$) related?

## Linear algebra – Computational complexity of calculating the trace of a matrix product under a certain structure

I have two problems with calculating a trace and some responses (possibly suboptimal). My question is about a potentially more efficient algorithm for everyone. (More interested in an answer to question 1.)

1. Let $$U, V$$ and $$F$$ to be three real matrices. The three matrices have a size $$d times r$$, with $$r ll d$$ (It is, $$U, V$$ and $$F$$ are “ great & # 39; & # 39;). I want to calculate $$mathrm {trace} (U V ^ top F F ^ top)$$. IT $$A = F ^ top U$$, $$B = V ^ top F$$ and the trace of $$AB$$ has complexity $$mathcal {O} (r ^ 2 d)$$. Is there a faster algorithm (taking into account $$r ll d$$)? Can we have $$mathcal {O} (r d)$$?

2. Let $$U, V$$ and $$M$$ to be three real matrices. $$U$$ and $$V$$ have the size $$d times r$$ (with $$r ll d$$), and $$M$$ is lower triangular (with positive elements in its diagonal) in size $$d times d$$. I want to calculate $$mathrm {trace} (U V ^ top M M ^ top)$$. The simple calculation algorithm $$A = M ^ top U$$, $$B = V ^ top M$$, then the trace of $$AB$$ has a complexity $$mathcal {O} (r d ^ 2)$$. Is there a faster algorithm (taking into account $$r ll d$$)?

If this question doesn't belong here, let me know! (If yes, where can I post it?)

Thank you!

## ag.algebraic geometry – Count the image of a variety map using the trace formula

Assume $$f: X to Y$$ is a finite map of varieties on a finite field $$mathbb F_q$$. Is there a buildable stall $$mathbb Q_ ell$$ sheaf $$mathscr F$$ sure $$Y$$ which counts the number of rational points of the form $$f (x)$$ for $$x$$ itself rational (like an application of the trace formula)?

Yes $$f$$ is closed, we can just use pushforward. On the other hand, even if $$X, Y$$ are the two field spectra, let's say of $$mathbb F_ {q ^ n}, mathbb F_q$$, so I'm not sure what we want.

Edit 1: It seems to me that it might be easier to count $$deg (f)$$ times the number of points and that's fine too. For example, the number of squares in $$mathbb P ^ 1$$ East $$(q-1) / 2 + 2$$ but since the eigenvalues ​​are algebraic integers, we cannot obtain a factor of $$q / 2$$ by cohomology calculations. But if we multiply by $$2$$, it is possible.

Edit 2: Watch the example of $$(.) ^ ell: mathbb G_m to mathbb G_m$$, number of $$mathbb F_ {q ^ r}$$ points depends on whether or not $$ell | q ^ r-1$$ (it's either $$(q ^ r-1)$$ or $$(q ^ r-1) ell$$ after multiplication by degree). If we assume that $$q$$ is a generator for $$( mathbb Z / ell) ^ times$$, then it seems that on cohomology with compact support of degree zero, we want the eigenvalues ​​to be $$q zeta _ { ell-1} ^ {k}$$ for $$1 leq k leq ell$$ and the same for the degree $$0$$.

## dnd 5th – Are there any magic items in 5th that will launch Pass Without Trace?

To complete MivaScott's answer, a Spell Storage Ring could serve your purpose.

Spell Storage Ring

This ring stores the spells that are cast there, holding them until the listening wearer uses them. The ring can store up to 5 spell levels at a time. Once found, it contains 1d6 – 1 levels of stored spells chosen by the DM.

Any creature can cast a spell from 1st to 5th level in the ring by touching the ring while the spell is cast. The spell has no effect, except to be stored in the ring. If the ring cannot contain the spell, the spell is spent with no effect. The level of the location used to cast the spell determines the amount of space it uses.

By wearing this ring, you can cast any spell it contains. The spell uses location level, spell save DD, spell attack bonus, and the spellcasting ability of the original spellcaster, but is otherwise treated as if you cast the spell. The cast spell of the ring is no longer stored there, freeing up space.

Thus, your DM could create a Ring of Spell Storing with Pass Without Trace stored in it, allowing you to cast the spell. The ring can store a maximum of two Pass Without Trace spells and another level one.

If you need a refill, your DM could allow an NPC in town to cast this spell for you to store it.

## labeling – Unexpected warnings issued by ListPointPlot3D when trying to trace with labels

I made a very simple layout with the code below:

``````ListPointPlot3D[{{-1.7320508075688776`, -4.`, -1.1796`},
{-1.7320508075688772`, -3.`, -1.1796`}, {-1.7320508075688776`, -3.`,
1.1796`}, {-1.7320508075688772`, -2.`, 1.1796`}} -> {1, 2, 3, 4}]
``````

who returned the desired figure, but with mysterious warnings, I couldn't understand.

Does anyone know what happened? BTW, I'm using Windows 10 + V 12.1.

## ct.category theory – Denis trace map for stable \$ infty \$ -category, naively

I'm trying to get more information on upper K theory, Hochschild homology and the trace map between thinking about these objects from an informal $$infty$$– strategic perspective, instead of using very precise and concrete definitions:

Let $$A$$ to be a little stable $$infty$$-Category. If my understanding is correct, Hochschild's topological homology $$A$$ can be described as the co-end:

$$THH (A): = int ^ A text {Hom} (a, a)$$

Informally, this means that a spectrum map $$THH (A) rightarrow X$$ defines a $$X$$-trace evaluated for arrows $$A$$, i.e. spectrum maps $$text {Tr} _a: text {Hom} (a, a) to X$$ which satisfies a certain coherence relation, the first corresponding to "$$text {Tr} (ab) = text {Tr} (ba)$$", and higher being the coherence conditions for the cyclic permutation trace of the composite of a $$n$$– cycle of arrows in $$A$$. Therefore $$THH (A)$$ is the target of the universal trace map for $$A$$.

On the other hand, the (connective) $$K$$-the theory of $$A$$ can be considered (for example, from the construction of Waldhausen) the target of the universal card "Characteristic of Euler" for $$A$$, in the sense that a spectrum map $$K (A) to X$$ corresponds to a characteristic of Euler $$chi (a) in X$$ for each $$a in A$$, so if $$a à b à c$$ is a sequence of fibers so we have an equivalence $$chi (b) simeq chi (a) + chi (c)$$ also subject to higher consistency conditions.

So it seems natural that Denis' trace map $$K (A) to THH (A)$$ in this perspective, should correspond to the map defined informally:

$$begin {array} {ccc} K (A) & to & THH (A) \ chi (a) & mapsto & text {Tr} ( text {Id} _a) end {array}$$

To show that this defines cards, we have to show that if $$a à b à c$$ is a sequence of fibers $$A$$ then, for any trace function like above, we can construct an equivalence $$text {Tr} ( text {Id} _b) simeq text {Tr} ( text {Id} _a) + text {Tr} ( text {Id} _c)$$

Of course, for complete proof that this card is well defined as a specter card (or rather $$E_ infty$$-algebras by identifying the connective spectra with grouplike $$E_ infty$$-algebra) we would also need to deal with "higher coherence conditions", and show that this induces a groulike map $$E_ infty$$ algebra, dealing with a higher consistency condition and thus a. But I focus on the first condition because it is the first that I do not understand.

Question: Can we give formal / elementary proof that a trace card like above is automatically accompanied by equivalences $$text {Tr} ( text {Id} _b) simeq text {Tr} ( text {Id} _a) + text {Tr} ( text {Id} _c)$$ for each fiber sequence $$a à b à c$$.

I think I know how to prove it using a deeper theorem, for example the additivity of $$THH$$, but I'm really interested in some direct elementary proof of this.

To give an idea of ​​the type of argument that I accept as an answer, it is easy to prove formally that if $$b = a oplus c$$ then $$text {Tr} ( text {Id} _b) = text {Tr} ( text {Id} _a) + text {Tr} ( text {Id} _c)$$. Indeed, as $$a$$ and $$c$$ are retracted from $$b = a oplus c$$, with idempotent $$P_a, P_c: b to b$$. The trace property shows that:

$$text {Tr} (P_a) = text {Tr} (i_a p_a) = text {Tr} (p_a i_a) = text {Tr} ( text {Id} _a)$$

and $$text {Id} _b = P_a + P_c$$ therefore:
$$text {Tr} (Id_b) = text {Tr} (P_a) + text {Tr} (P_c) = text {Tr} ( text {Id} _a) + text {Tr} ( text {Id} _c)$$

## tracing – Error when using the Trace command: cannot specify the y range window without using PlotRange instead of just typing the option {y, ymin, ymax}

I had a problem with Mathematica.

I got this error message when i set the y-value range of a graph using the `Plot` command (I will use `f(x)=x^2` for example):

Contribution:

`Plot(x^2, {x, -100, 100}, {y, 0, 100})`

Production:

``````Plot::nonopt: Options expected (instead of {y,0,100}) beyond position 2 in Plot(x^2,{x,-100,100},{y,0,100}). An option must be a rule or a list of rules.
``````

I think it may be because I have an older MacBook Pro. This is due to the fact that I am the only one in class 1) with this problem and 2) who has a Mac. I'll start with the specifications:

Wolfram Mathematica 12 Student Edition

Mathematica version number: 12.0.0.0

MacBook Pro (15 inch, mid-2010)

Platform: Mac OS X x86 (64-bit)

There is a slight workaround that allows me to specify the range of y values, but I get the impression that the output might be wrong:

``````Plot(x^2, {x, -1, 1}, PlotRange -> {{-.5, .5}, {0, 1.5}})
``````

It's strange, however. The first declaration of the x range option to the Plot command did not seem to change the graph at first. However, when I change `{x,-1,1}` at `{x,-1000,1000}` for example, the graph has changed:

``````Plot(x^2, {x, -1000, 1000}, PlotRange -> {{-.5, .5}, {0, 1.5}})
``````

Ope! I just realized how I can use `PlotRange` to get a safe exit, I can trust:

`Plot(x^2, {x, -.5, .5}, PlotRange -> {0, 1.5})`

Enter this command with the `PlotRange` I have no problem with this option, although I still don't know why other end users can just type easily:

`Plot(x^2, {x, -.5, 5}, {y, 0, 1.5})`

``````Plot::nonopt: Options expected (instead of {y,0,1.5}) beyond position 2 in Plot(x^2,{x,-0.5,5},{y,0,1.5}). An option must be a rule or a list of rules.
``````

I also tried to join the options with wavy braces:

`Plot(x^2, {{x, -.5, 5}, {y, 0, 1.5}})`

``````Plot::nonopt: Options expected (instead of {y,0,1.5}) beyond position 2 in Plot(x^2,{x,-0.5,5},{y,0,1.5}). An option must be a rule or a list of rules.
``````

I am ready to solve this problem – and again – thank you for your time! I hope it's just a quick fix like missing a special character or something …

minnesnowta11