## Plotting Integral of Exponential functions

I am trying to Plot an integral equation that involves exponential function. My code is as follow,

L[\[Alpha]_] :=    NIntegrate[    1/(k + I*0.1) (     Exp[I*k*x] (Exp[Sqrt[k^2 + \[Alpha]/w^2]*w] - 1) (Exp[k*w] - 1 +         I*0.1) Sqrt[      k^2 + \[Alpha]/       w^2])/((Sqrt[k^2 + \[Alpha]/w^2] + k) (Exp[         Sqrt[k^2 + \[Alpha]/w^2]*w - Exp[k*w]]) + (Sqrt[         k^2 + \[Alpha]/w^2] -          k) (Exp[(k + Sqrt[k^2 + \[Alpha]/w^2]) w] -          1)), {k, -100, 100}]; Plot[{Re[L[10]], Re[L[100]], Re[L[500]]}, {x, -0.45, 0.45},   PlotRange -> Full].  

But this integral gives a lot of oscillations which it should not. This is fig 2 in this article “https://arxiv.org/pdf/1508.00836.pdf” that I am trying to plot. Any help will be highly appreciated.

## TQBF PSPACE-COMPLETE : Why this algorithm is exponential but Savitch’s not?

So this is a question pertaining to the proof for $$PSPACE-COMPLETE$$ (for TQBF for example). The idea is to first prove the $$L$$ $$is$$ $$PSPACE$$(easy part) and next is to prove $$PSPACE-COMPLETE$$. The latter requires demonstrating an algorithm which computes the L in polynomial space. This is usually achieved by having recursive calls such that is re-used.

In TBQF proof, the equation $$\phi_{i+1}(A,B)$$= $$\exists Z [\phi_{i+1}(A,Z) \land \phi_{i+1}(Z,B) ]$$ ($$Z$$ is mid-point )is default recursive relation for computing TBQF truth. In any standard proof, it is said that $$\phi_{i+1}(A,B)$$ is computed two times and for $$m$$ nodes, this formula explodes hence, other recursive-relation should be used to bound.

However in Savitch’s proof, the recursive relation is $$Path(a,b,t)$$ = $$Path(a,mid,t-1)$$ AND $$Path(mid,b,t-1)$$ accepts then ACCEPT. In proof, it is stated that this relation reuses-spaces.

My Question is Why in TBQF relation space explodes while in Path, it is reused? Both of these relations looks more or less same to me because both refers to i-1 instances and will need space to store them?.

## What is the complexity class of exponential parallelism?

Consider the class of problems that can be computed when you have access to exponentially many processors working in parallel.

How does one capture that in a proper formalism? Is there some literature on it already? Would it be true that such a class would be a superset of BQP?

## Positive semi-definite block diagonal covariance matrix with exponential decay

I am implementing Kalman filtering in R. Part of the problem involves generating a really huge error covariance block-diagonal matrix (dim: 18000 rows x 18000 columns = 324,000,000 entries). We denote this matrix Q. This Q matrix is multiplied by another huge rectangular matrix called the linear operator, denoted by H.

I am able to construct these matrices but it takes a lot of memory and hangs my computer. I am looking at ways to make my code efficient or do the matrix multiplications without actually creating the matrices exclusively.

library(lattice) library(Matrix) library(ggplot2)  nrows <- 125  ncols <- 172  p <- ncols*nrows  #--------------------------------------------------------------# # Compute Qf.OSI, the "constant" model error covariance matrix # #--------------------------------------------------------------#    Qvariance <- 1   Qrho <- 0.8    Q <- matrix(0, p, p)     for (alpha in 1:p)   {     JJ <- (alpha - 1) %% nrows + 1     II <- ((alpha - JJ)/ncols) + 1     #print(paste(II, JJ))      for (beta in alpha:p)     {       LL <- (beta - 1) %% nrows + 1       KK <- ((beta - LL)/ncols) + 1        d <- sqrt((LL - JJ)^2 + (KK - II)^2)       #print(paste(II, JJ, KK, LL, "d = ", d))        Q[alpha, beta] <-  Q[beta, alpha] <-  Qvariance*(Qrho^d)     }   }     # dn <- (det(Q))^(1/p)   # print(dn)    # Determinant of Q is 0   # Sum of the eigen values of Q is equal to p    #-------------------------------------------#   # Create a block-diagonal covariance matrix #   #-------------------------------------------#    Qf.OSI <- as.matrix(bdiag(Q,Q))    print(paste("Dimension of the forecast error covariance matrix, Qf.OSI:")); print(dim(Qf.OSI)) 

It takes a long time to create the matrix Qf.OSI at the first place. Then I am looking at pre- and post-multiplying Qf.OSI with a linear operator matrix, H, which is of dimension 48 x 18000. The resulting HQf.OSIHt is finally a 48×48 matrix. What is an efficient way to generate the Q matrix? The above form for Q matrix is one of many in the literature. In the below image you will see yet another form for Q (called the Balgovind form) which I haven’t implemented but I assume is equally time consuming to generate the matrix in R.

## Exponential Deconvolution Using the First Derivative

There is an interesting observation using the first derivative to deconvolve an exponentially modified Gaussian:

The animation is here, https://terpconnect.umd.edu/~toh/spectrum/SymmetricalizationAnimation.gif

The main idea is that if we have an Exponentially Modified Gaussian (EMG) function, and we add a small fraction of first derivative to the original EMG, it results in recovering the original Gaussian while preserving the original area. The constant multiplier is the 1/time constant of the EMG. This is a very useful property.

Has anyone seen this deconvoluting property of the first derivative mentioned elsewhere in mathematical literature? An early reference from the 1960s from a Chemistry paper shows a picture a similar picture. This observation was just by chance, I am looking for a fundamental connection and if the first derivative can be used to deconvolute other types of convolutions besides the exponential ones.

Thanks.

Ref: J. W., and Charles N. Reilley. “De-tailing and sharpening of response peaks in gas chromatography.” Analytical Chemistry 37, (1965), 626-630.

## Formula for exponential integral over a cone

While reading ‘Computing the Volume, Counting Integral points, and Exponential Sums’ by A. Barvinok (1993), I came across the following:

“Moreover, let $$K$$ be the conic hull of linearly independent vectors $$u_{1}, …, u_{n}$$ so $$K = co(u_{1}, …, u_{n})$$. Denote by $$K^{*} = \{x\in \mathbb{R}^{n}: \le 0 \text{ for all } y\in K\}$$ the polar of K. Then for all $$c \in \text{Int} K^{*}$$ we have

$$$$\int_{K}exp()dx = |u_{1} \land … \land u_{n}|\prod_{i=1}^{n}<-c_{i}, u_{i}>^{-1}$$$$

To obtain the last formula we have to apply a suitable linear transformation to the previous integral. “

I have tried proving this but I can’t find relevant links to help me. Also, I’m unsure what $$|u_{1} \land … \land u_{n}|$$ is supposed to mean. Would greatly appreciate if someone could point me to a relevant resource or provide proof. Thanks!

## Exponential decay of a $J$-holomorphic map on a long cylinder

Suppose $$(X,\omega,J)$$ is a closed symplectic manifold with a compatible almost complex structure. The fact below follows from McDuff-Salamon’s book on $$J$$-holomorphic curves (specifically, Lemma 4.7.3).

Given $$0<\mu< 1$$, there exist constants $$0 and $$\hbar>0$$ such that the following property holds. Given a $$J$$-holomorphic map $$u:(-R-1,R+1)\times S^1\to X$$ with energy $$E(u)<\hbar$$ defined on an annulus (with $$R>0$$), we have the exponential decay estimates

(1) $$E(u|_{[-R+T,R-T]\times S^1})\le C^2e^{-2\mu T}E(u)$$

(2) $$\sup_{[-R+T,R-T]\times S^1}\|du\|\le Ce^{-\mu T}\sqrt{E(u)}$$

for all $$0\le T\le R$$. Here, we take $$S^1 = \mathbb R/2\pi\mathbb Z$$ and use the standard flat metric on the cylinder $$\mathbb R\times S^1$$ and the metric on $$X$$ to measure the norm $$\|du\|$$.

Now, if $$J$$ were integrable, we can actually improve this estimate further in the following way: at the expense of decreasing $$\hbar$$ and increasing $$C$$, we can actually take $$\mu=1$$ in (1) and (2) above. The idea would be to use (2) to deduce that $$u|_{[-R,R]\times S^1}$$ actually maps into a complex coordinate neighborhood on $$X$$ where we can use the Fourier expansion of $$u$$ along the cylinder $$[-R,R]\times S^1$$ to obtain the desired estimate.

I would like to know: is it possible to improve the estimate to $$\mu=1$$ also in the case when $$J$$ is not integrable? If so, a proof with some details or a reference would be appreciated. If not, what is the reason and is it possible to come up with a (counter-)example to illustrate this?

## Can any NP-Complete Problem be solved using at most polynomial space (but while using exponential time?)

I read about NPC and its relationship to PSPACE and I wish to know whether NPC problems can be deterministicly solved using an algorithm with worst case polynomial space requirement, but potentially taking exponential time (2^P(n) where P is polynomial).

Moreover, can it be generalised to EXPTIME in general?

The reason I am asking this is that I wrote some programs to solve degenerate cases of an NPC problem, and they can consume very large amounts of RAM for hard instances, and I wonder if there is a better way. For reference see https://fc-solve.shlomifish.org/faq.html .

## Exponential search worst-case performance

I’m learning about Exponential search and I keep reading that the worst-case performance is $$log_2(i)$$ where $$i$$ is the searched index.

I tried with an array containing $$1073741824$$ elements $$(1024*1024*1024)$$ and I searched for the $$1073741823$$th element (which is just $$n-1$$).

Therefore, I used the exponential search to get the range containing $$i$$ (which was $$[536870912 ; 1073741824]$$ and it took $$log_2(n)$$ to find. Then it took another $$log_2(n/2)$$ to find the right index using Binary search in this range only.

Am I doing something wrong? Or is the complexity of the worst-case $$2*log_2(n)-1$$ ?

## Compare my software’s representation of exponential numbers and 0?

Suppose I have a real number $$x=\sum_{i=1}^n a_i e^{\lambda_i}$$ where $$a_i,\lambda_i$$s are complex algebraic numbers.

Is there an algorithm to determine whether it is greater than 0 or less than 0?