text:
# Learning UE4 Niagara

I’m working through some tutorials on how to use the video FX module in Unreal Engine. Here, I’ve outlined a static mesh of my favorite element with glowing embers and added a very rudimentary lens flare.

# 2D, 3D, let’s explore molecular geometry

It should go without saying that I am a fan of incorporating digital fabrication tools into chemistry instruction. A while back, Scott McIndoe and his colleagues published a paper in the Journal of Chemical Education on using laser cut models to teach molecular geometry. At first, it seems counter-intuitive to use a 2D design process to teach 3D topics. On top of that, if you have access to a laser cutter, you probably have access to a 3D printer, so why not just use the 3D printer?

3D printing is great for rapid prototyping; however, when you want to scale up production – say, for example, you want models for a classroom of 100 students – then 3D printing is not a great solution. Laser cutting is one of the fastest, easy-access digital fabrication tools available, and build times can be drastically reduced if a laser-cutter solution is available.

Then again, one needs access to a laser cutter. And that is something I (sadly) do not have at present.

So I took Scott’s design and adapted it to 3D printing. Yes, his paper provides some STLs to print, but if you want to create different sizes, then you have to do some scaling. I wanted a parametric approach, so I resorted to OpenSCAD to recreate his design. The code is below, in case you are interested.

Why would you want to use this? I don’t know, perhaps you want to make super thin molecules or super small molecules. All I know is, now you can.

/*
Origin: @mcindoe
Reference: https://pubs.acs.org/doi/10.1021/acs.jchemed.8b00553
*/

ca_d = 12; // diameter of central atom
ca_g = 5;  // length of the gap in the central atom
oa_d = 10; // diameter of the outer atom
bond_w = 5; // width of the bond
bond_l = 25; // length of the bond
th = 0.2+3*0.35; // piece thickness (my printer is set to 0.2 mm first layer and 0.35 subsequent layers)
allowance = 0.05; // adjust gap size for tighter/looser fit, YMMV
\$fn = 25; // Keeps round parts round

/*
n is the number of bonds (1, 2 or 3).  Setting tet to true will force a 109.5 degree bond angle.  NOTE: you shouldn't call this function, use makepiece
*/
module makebonds(n,tet=false) {
for(i =[1:n]) {
rot = tet?125.25+(i-1)*109.5:i*360/(n+1);
rotate(rot,[0,0,1])
translate([bond_l,0,0])
union(){
circle(d=oa_d);
translate([-bond_l,-bond_w/2,0])square([bond_l,bond_w]);
}
}
}

/*
n is the number of bonds (1-3).  Setting tet to true will force a 109.5 degree bond angle
*/
module makepiece(n,tet=false) {
rot = (tet && (n==1))?70.5:0;
linear_extrude(th)difference(){
union(){
circle(d=ca_d);
makebonds(n,tet);
}
rotate(rot,[0,0,1])translate([ca_d/2-ca_g,-(th+allowance)/2,0])square([ca_g,th+allowance]);
}
}

// Example usage
for(i=[ [-10,-20,0], [10,20,0] ]){
translate(i){
makepiece(3);
translate([40,0,0])makepiece(1);
translate([-40,0,0])makepiece(2);
}
}

# Final project of the summer

So I’ve wrapped up my final project of the summer. During the pandemic, I’ve been brushing up on my electronics and learning how to use the Unreal Engine to create video games. I wanted to see if I could tie those two ideas together. Watch the video below to see how it’s possible to create a video game that requires you to know what is going on in the physical world. In this case, the color of an LED connected to a microcontroller informs the gamer which colored sphere to shoot.

# UE4: Editor assist

I’m continuing to wrap up some summer projects before school starts. I’ve been learning how to create games in Unreal Engine over the past few weeks. One tool that I’ve found handy has been the ability to use Blueprints while in the editor to make life easier. This is one such example, where I use Blueprints to place an actor where the in-editor camera is located.

# Web-enabled games

Summer is almost over, and I am cranking through a bunch of projects I had wanted to complete. One of them was to spend as much time as possible learning how to make video games.

I don’t have any delusions of grandeur; however, I do have a fantasy of being able to combine game making with electronics and making in order to design a game that interacts with the physical world. In order to do that, I need a way for the game to communicate with the physical world.

In a previous post, I highlighted some of the things I learned during my first 100 hours with the Unreal Engine. That was enough time (with the right tutorials) to build a simple game called Chicken in the Lab where your a Chicken, in a lab, collecting elements. The trick is that you have to collect them in order of atomic number, so it can be a bit of a challenge.

With some general knowledge on game design using the Unreal Engine, I wanted to learn how a game could interact with the world outside of the game. There’s a plugin called VaREST which allows the game to communicate via REST API calls; however, the documentation on the plugin is virtually nonexistent. I decided to demonstrate how I got it working primarily to have all of this information in one place when I need it again, but perhaps it will be of use to someone.


text:
# Difference-in-difference-in-differences estimator

Two towns, $X$ and $Y$. In each town:

• Pool cross-sections of male and female hourly wages, one from the year before a wage-discrimination policy took effect and one from the year after.

Consider the following model:

• $wage_i=\beta_0+\beta_1after_i+\beta_2female_i+\beta_3X_i +\beta_4after_iX_i +\beta_5after_ifemale_i +\beta_6X_ifemale_i +\beta_7after_ifemale_iX_i+u_i$

where

• $after_i=1$ if date after gender-wage discrimination policy; $0$ if date before gender-wage discrimination policy. The gender wage discrimination policy only applies to females in town $X$.

• $female_i=1$ if female; $0$ if male.

• $X_i=1$ if town $X$; $0$ if town $Y$.

The treatment group is females from town $X$.

Does this model measure the effect of a gender-wage discrimination policy on the average wage of women relative to men in town $X$ compared to the average wage of women relative to men in town $Y$?

In this model, the affect of the policy is captured by $\beta_7$:the 'difference-in-difference-in-differences' estimator.

For $\hat{\beta_7}$ to be interpreted as the causal effect of the policy on wages in town $X$, $E(u_i|female_i, after_i, X_i)=0$ has to be assumed.

Does this mean that once gender, date of policy, and town have been controlled for, there exist no unobserved factors which change over both town and time that affect wages?

• When you say "pooled cross section", is this what you actually mean or do you mean that you have panel data? This can make a difference for the robustness of your estimation strategy. See the discussion here. – Andy May 18 '14 at 13:24

What you propose here is actually difference in difference in differences (DDD) instead of the usual difference in differences (see these lecture notes by Imbens and Wooldridge (2007) on the first two pages). This method can potentially account for the unobserved trends in wages of women across your two towns and the wage changes of both male and female workers in the treatment town. In this sense DDD is a more robust approach to the DD approach you had in mind previously.

Let $\overline{y}$ be average income and index periods $1,2$, females and men $F,M$, and towns $X,Y$, then your DDD coefficient gives you \begin{align} \widehat{\beta}_7 &= (\overline{y}_{X,F,2} - \overline{y}_{X,F,1}) \enspace\quad \text{the time change in \overline{y} for women in town X} \newline &- (\overline{y}_{Y,F,2}-\overline{y}_{Y,F,1}) \qquad \text{the time change in \overline{y} for women in town Y} \newline &-(\overline{y}_{X,M,2}-\overline{y}_{X,M,1}) \enspace\quad \text{the time change in \overline{y} for men in town X} \end{align} so your treatment effect will still be the effect of the policy on female wages relative to male wages in town $X$. The second term subtracts the potential wage trend of females that has nothing to do with the policy. This relates to what I mentioned in my response to your other question that you would need to know the common trends between male and female wages. This was to exclude the possibility that male and female wages are subject to systematically different changes that have nothing to do with the policy. Having untreated females in another town allows to you take out this potential female wage trend that might be different from the male wage trend.
Read the Imbens and Wooldridge lecure very carefully because they do a great job on explaining this (especially see page 2 after equation 1.4).

Even though this method is more robust, it does not mean that there is nothing left which can bias your estimates. If there is another policy at the same time, for instance a law for mothercare that affects female wages, this might still be picked up by your regression. In your work it will be your job to show that nothing else is going on in town $X$ besides your discrimination policy.

• in the DDD estimate (beta7_hat) you need to add time change for men in town Y as well. – Seyhan Erden Jun 5 '17 at 13:46
• good catch Seyhan! you do need to add time change for men in town Y as well, Imbens and Woodridge kinda omit it. – wise Dec 23 '17 at 18:33
• This threw me for a loop as well. Perhaps that we should update the answer as well as the links to the notes. In a different set of lectures, Wooldridge (2011) includes the time change for men in town Y. I think there was just a mistake in the Imbens and Wooldridge (2007) notes. Some discussion of how to reconcile the intuition in the the Imbens and Wooldridge (2007) and the Wooldridge (2011) lecture notes may be helpful as well. Here are the Wooldridge 2011 notes: eief.it/files/2011/10/slides_6_diffindiffs.pdf – chandler Feb 16 '18 at 17:32
• what the above discussion (whether or not to add time change for men in town Y) is referred to equation 1.4 in this note(nber.org/WNE/lect_10_diffindiffs.pdf) – Jason Goal Apr 19 at 11:43
• @Andy, Hi Andy, good to see you here, you helped me a lot on the DID method ever since my first touch of this model. Thanks a lot. Concern the heterogeneity of treatment effect, is that possible to replace the binary variable discussed here (gender) with a continuous variable like income before treatment? – Jason Goal Apr 19 at 11:47

## protected by Community♦Dec 23 '17 at 18:49

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text:
## Algebra 2 (1st Edition)

$4! = 4*3*2*1 = 12*2 = 24$ as a factorial signifies the multiplication all natural numbers less than or equal to the argument.


text:
# What does fractional notation represent in molecular formulas such as SiO4/2 or H2SiO2/2?

I'm going through some papers about HSQ (hydrogen silsesquioxane) and keep seeing this fractional notation being used: HSiO3/2 , SiO4/2 , H2SiO2/2 , H3SiO1/2.

I understand that the proper formula for HSQ is [HSiO3/2]2n. Because we can't have fractional atoms, it's always multiplied by an even number and as such the fraction is never a consideration. But then why are the fractions maintained in the other compounds, and not reduced to SiO2 and H2SiO ?

Does keeping the fraction confer some structural information ? I've tried to find out what this notation means, but could not find a proper explanation.

Edit: What do the numerator and denominators mean, formally, in those formulas ? If it's that the oxigen atoms share outside bonds, why not write the compound as SiO4−4 ? If the numerator is the number of atoms, what does the denominator represent ? And is it always /2 ? Can we have /x , where x is a different integer ? I've never seen this type of notation before. Does the denominator mean that the valence electrons are halved ?

• Well, yes, it does confer certain structural information. You don't have an entity SiO2, for example. What you have is an entity SiO4 in which all oxygen atoms are shared with others. May 13 at 6:34
• So then what does the /2 in $SiO_{4/2}$ mean ? I've never seen the fractional notation before. What is it telling us ? That all of the oxigen atoms are shared ? Why "4/2" then if they all share outside bonds ? Why not write it as $SiO_{4}^{−4}$ ? If the numerator is the number of atoms, what does the denominator represent ? Right until now I've only seen whole numbers in the subscripts, not fractions. Just trying to understand what it represents formally. And is it always /2 ? Can we have /x , where x is a different integer ? May 13 at 12:01
• 4 stands for four oxygens, 2 means that each oxygen is shared between two SiO4 entities, and we don't write $\ce{SiO4^4-}$ because that's a thing we don't have here either. (We may have it elsewhere, but that's another story.) May 13 at 12:16
• Indeed, you'd better accept it in a more hand-wavy and less formal way. Even if there is a formal definition out there, this notation is far from being used universally. OK, SiO4/2 means that each oxygen atom is shared between two entities which do not have to be similar. Say, between SiO4/2 and HSiO3/2. May 13 at 13:34
• The terminology of metal clusters and ligands... well, I guess it can technically be used here, but then it doesn't sit well with writing "clusters" by parts. When you call your "ligand" $\mu_2$, you imply that it is $\mu_2$ with respect to the whole "cluster". May 14 at 7:36


text:
# 2.2.3.9.19 matrixbase::Find

## Description

Function to find all instances of a value in a matrix

## Syntax

int Find( vector<uint> & vecR, vector<uint> & vecC, double dLower, double dUpper = _ONAN, int nPrecision = 8, int r1 = 0, int c1 = 0, int r2 = -1, int c2 = -1 )

## Parameters

vecR
[output] row indices of all cells in matData where value was found.
vecC
[output] column indices of all cells in matData where value was found.
dLower
[input] the lower bound for the value to be found. Set this to exact value when you just want to search for a single value and not use bounds.
dUpper
[input] the upper bound for the value to be found. Leave it as default (_ONAN) if searching for only a single value.
nPrecision
[input] sets the level of precision to be used in the search. This parameter will be ignored if dUpper isn't the default one.
r1
[input] row index of cell in matData from where to start the search - leave as 0 to start from the first cell
c1
[input] column index of cell in matData from where to start the search - leave as 0 to start from the first cell
r2
[input] row index of cell in matData up to which the search should be done - leave as -1 to search to last cell
c2
[input] column index of cell in matData up to which the search should be done - leave as -1 to search to last cell

## Return

Return -1 if no values were found, otherwise return "n" where n is the number of values found.

## Examples

EX1

// Find matrix elements within a range
void matrixbase_Find_ex1()
{
int rc, ii,jj;

vector<uint> vr;
vector<uint> vc;
matrix<double> mat1 = {
{-3, -2, -1},
{99,  0,  0},
{ 1,  2,  3}
}

for(ii=0; ii<3; ii++)
for(jj=0; jj<3; jj++)
if(mat1[ii][jj]==99)
mat1[ii][jj]=NANUM;  // set row=ii,col=jj to NANUM
//  Input matrix is:
//        {-3, -2, -1}
//        {--,  0,  0}
//        { 1,  2,  3}

//   Find condition:  -1.0<= amd <=1.0
//   Output matrix to present the results (Found=1, Otherwise=0) is:
//        {0,  0,  1}
//        {0,  1,  1}
//        {1,  0,  0}

MatrixPage MatPg1;
MatPg1.Create("Origin");
MatrixLayer MatLy1 = MatPg1.Layers(0);
Matrix Mat1(MatLy1);
Mat1 = mat1;
printf("  Input matrix is %s.\n",Mat1.GetName());

MatrixPage MatPg2;
MatPg2.Create("Origin");
MatrixLayer MatLy2 = MatPg2.Layers(0);
Matrix Mat2(MatLy2);

rc=mat1.Find(vr, vc, -1, 1); // Demonstration of to Find to find elements of -1.0<=cell<=1.0

if(rc==-1)
printf("  Error: Find failed.\n");
else {
printf("  Results of Find are presented in %s. (Found=1, Otherwise=0)\n",Mat2.GetName());
printf("  Find condition: From:-1.0 To:1.0 (inclusive).\n");
printf("  Find found %d elements.\n",rc);

Mat2.SetSize(mat1.GetNumCols(),mat1.GetNumRows());
for(ii=0; ii<mat1.GetNumCols(); ii++)
for(jj=0; jj<mat1.GetNumRows(); jj++)
Mat2[ii][jj]=0;
printf("  Found Elements:\n");
for(ii=0; ii<vr.GetSize(); ii++) {
Mat2[vr[ii]][vc[ii]]=1;
printf("    #%d) Cell(%d,%d)=%g\n",ii+1,vr[ii]+1,vc[ii]+1,mat1[vr[ii]][vc[ii]]);
}
printf("  Note that the missing values are excluded from the result.\n");
}
}

## Remark

The data of NANUM can also be available in Find function.


text:
Timezone: »

Poster
Estimators for Multivariate Information Measures in General Probability Spaces
Arman Rahimzamani · Himanshu Asnani · Pramod Viswanath · Sreeram Kannan

Wed Dec 05 02:00 PM -- 04:00 PM (PST) @ Room 210 #80
Information theoretic quantities play an important role in various settings in machine learning, including causality testing, structure inference in graphical models, time-series problems, feature selection as well as in providing privacy guarantees. A key quantity of interest is the mutual information and generalizations thereof, including conditional mutual information, multivariate mutual information, total correlation and directed information. While the aforementioned information quantities are well defined in arbitrary probability spaces, existing estimators employ a $\Sigma H$ method, which can only work in purely discrete space or purely continuous case since entropy (or differential entropy) is well defined only in that regime. In this paper, we define a general graph divergence measure ($\mathbb{GDM}$), generalizing the aforementioned information measures and we construct a novel estimator via a coupling trick that directly estimates these multivariate information measures using the Radon-Nikodym derivative. These estimators are proven to be consistent in a general setting which includes several cases where the existing estimators fail, thus providing the only known estimators for the following settings: (1) the data has some discrete and some continuous valued components (2) some (or all) of the components themselves are discrete-continuous \textit{mixtures} (3) the data is real-valued but does not have a joint density on the entire space, rather is supported on a low-dimensional manifold. We show that our proposed estimators significantly outperform known estimators on synthetic and real datasets.


text:
# Lectures on Partial Differential Equations

Publisher:
Springer Verlag
Publication Date:
2004
Number of Pages:
157
Format:
Paperback
Series:
Universitext
Price:
49.95
ISBN:
3-540-40448-1
Category:
Textbook
[Reviewed by
Victor Shubov
, on
01/20/2001
]

This book is written by a mathematician who is famous both for his fundamental contributions to modern mathematics and as an author of several textbooks and monographs on Mathematical Methods in Classical Mechanics, Ordinary Differential Equations, Singularities of Differentiable Maps, and Topological Methods in Hydrodynamics. The book is based on a short course of lectures delivered to the third year mathematics students of the Independent University of Moscow in the fall semester of 1994.

This introduction to PDEs is highly unusual and is very different from numerous textbooks on the subject. The book is not designed to serve as a systematic textbook for, e.g., science and engineering students. The reason is twofold. First, the reader is assumed to have a background in abstract differential geometry. Namely, such notions as differential forms,jets, symplectic and contact structures, cotangent bundle, Riemannian manifold, projective space etc. are used from the very beginning of the text. (Some of these notions are briefly defined but the reader should definitely be familiar with them.) Secondly, only a number of selected topics is discussed. For example, the wave equation and separation of variables are considered only in the case of a one -dimensional vibrating string. The heat equation is not considered at all. Some topics, e.g., boundary value problems for the Laplace equation, are described in a short verbal paragraphs. In fact, several topics whose description in standard texts involves a lot of important formulas are described in short verbal paragraphs. These descriptions are always very precise and go directly to the heart of the topic. However, the reader must definitely have sufficient mathematical maturity and technical skills to translate these short explanations to the language of formulas.

The topics that are selected for discussion are treated with great elegance and in a nonstandard manner, emphasizing both physical intuition and geometric insight. Some parts of the lectures contain material whose accessible exposition can hardly be found in any any other place. An example of this is an appendix devoted to topological content of the multifield representation of spherical functions (i. e., the fact that spherical functions can be obtained by calculating all partial derivatives of the Coulomb potential). The book contains many nontrivial problems.

The book can serve as a nonstandard, geometrically motivated introduction to PDEs for students having some background in pure abstract mathematics. It will also be of definite interest to specialists since it provides a fresh unusual approach to some standard topics. It is, probably, worth mentioning that the introduction contains some general philosophical views of the author on the subject of PDEs and modern mathematics as a whole and will be of interest to a broad mathematical audience.

Victor Shubov is Visiting Professor of Mathematics at Colby College.

Contents
Preface to the Second Russian Edition . . . . . . . . . . . . . . . . . . . . . . . . . V
1. The General Theory for One First-Order Equation . . . . . . . . 1
Literature . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 10
2. The General Theory for One First-Order Equation
(Continued) . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 11
Literature . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 20
3. Huygens' Principle in the Theory of Wave Propagation . . . . 21
4. The Vibrating String (d'Alembert's Method) . . . . . . . . . . . . . . 27
4.1. The General Solution . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 27
4.2. Boundary-Value Problems and the Cauchy Problem . . . . . . . . . 28
4.3. The Cauchy Problem for an Infinite String. d'Alembert's
Formula . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 29
4.4. The Semi-Infinite String . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 31
4.5. The Finite String. Resonance . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 31
4.6. The Fourier Method . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 32
5. The Fourier Method (for the Vibrating String) . . . . . . . . . . . . 35
5.1. Solution of the Problem in the Space of Trigonometric
Polynomials . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 35
5.2. A Digression . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 36
5.3. Formulas for Solving the Problem of Section 5.1 . . . . . . . . . . . . . 36
5.4. The General Case . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 36
5.5. Fourier Series . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 37
5.6. Convergence of Fourier Series . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 37
5.7. Gibbs' Phenomenon . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 39
X Contents
6. The Theory of Oscillations. The Variational Principle . . . . . . 41
Literature . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 50
7. The Theory of Oscillations. The Variational Principle
(Continued) . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 51
8. Properties of Harmonic Functions . . . . . . . . . . . . . . . . . . . . . . . . . 65
8.1. Consequences of the Mean-Value Theorem . . . . . . . . . . . . . . . . . . 67
8.2. The Mean-Value Theorem in the Multidimensional Case . . . . . . 73
9. The Fundamental Solution for the Laplacian. Potentials . . . 77
9.1. Examples and Properties . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 78
9.2. A Digression. The Principle of Superposition . . . . . . . . . . . . . . . . 79
9.3. Appendix. An Estimate of the Single-Layer Potential . . . . . . . . 89
10. The Double-Layer Potential . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 93
10.1.Properties of the Double-Layer Potential . . . . . . . . . . . . . . . . . . . 94
11. Spherical Functions. Maxwell's Theorem. The Removable
Singularities Theorem . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 105
12. Boundary-Value Problems for Laplace's Equation. Theory
of Linear Equations and Systems . . . . . . . . . . . . . . . . . . . . . . . . . . 121
12.1. Four Boundary-Value Problems for Laplace's Equation . . . . . . . 121
12.2.Existence and Uniqueness of Solutions . . . . . . . . . . . . . . . . . . . . . 125
12.3.Linear Partial Differential Equations and Their Symbols . . . . . . 127
A. The Topological Content of Maxwell's Theorem on the
Multifield Representation of Spherical Functions. . . . . . . . . . . 135
A.1. The Basic Spaces and Groups . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 136
A.2. Some Theorems of Real Algebraic Geometry . . . . . . . . . . . . . . . . 137
A.3. From Algebraic Geometry to Spherical Functions . . . . . . . . . . . . 139
A.4. Explicit Formulas . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 141
A.5. Maxwell's Theorem and CP2/conj ~ S4 . . . . . . . . . . . . . . . . . . . . 144
A.6. The History of Maxwell's Theorem . . . . . . . . . . . . . . . . . . . . . . . . 145
Literature . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 146
B. Problems . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 149
B.1. Material from the Seminars . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 149
B.2. Written Examination Problems. . . . . . . . . . . . . . . . . . . . . . . . . . . . 156


text:
Warning

# Homework 06 - Panorama

### Input Images

Consider the images were captured with the following EXIF info (focal length is in milimeters):

Tag Name Value
0xa002 ExifImageWidth 2400
0xa003 ExifImageHeight 1800
0xa20e FocalPlaneXResolution 2160000/225
0xa20f FocalPlaneYResolution 1611200/168
0xa210 FocalPlaneResolutionUnit inch
0x920a FocalLength 7400/1000

The images were then resized.

 07 06 05 04 - reference 03 02 01

### Steps

1. Manually create 10 image matches (point correspondences) between every pair of adjacent images. Store in the cell matrix u such that u{i,j} are points in the i-th image and u{j,i} are corresponding points in the j-th image. Save in 06_data.mat
2. Find homographies H_ij for every pair of adjacent images by the same method (optimizing 4 over 10) as in HW-05; use your u2h_optim function for each homography.
3. Draw 6 histograms of transfer errors of all correspondences into a single image. Use different colour for every histogram, export as 06_histograms.pdf.
4. Compute homographies H_i4 for every i ∈ <1,7> that maps the images above into the reference image 04 (thus H_44 is identity). Then construct inverse homographies H_4i.
5. Store all homographies into the cell matrix H, where H{i,j} is the homography from i to j. Save in06_data.mat.
6. Plot the image borders of the images 02 to 06 transformed to the image plane of the image 04. Export as 06_borders.pdf.
7. Construct a projective panoramic image from the images 03, 04, and 05, in the image plane of the image 04. Save as 06_panorama.png.
8. Construct calibration matrix K using the actual image size and the original EXIF. All the images share the same calibration. Store it in 06_data.mat
9. Establish a transformation between each image plane and the coordinate system of the cylinder, described below.
10. Plot the image borders of all the images mapped onto the cylinder (06_borders_c.pdf).
11. Construct a panoramic image from all the images mapped onto the cylinder (06_panorama_c.png).
12. Optionally, construct a cylindrical panorama from a set of your images (total pan at least 180 degrees) (06_panorama_c_own.png). Credited by at most 2 bonus points. In this case submit the input images as well (06_image01.jpg, etc ).
Example of histogram of errors
Example of image borders (origin not shifted)
Example of projective panorama (images 3-4-5)
Example of cylindrical panorama (all images)

### The cylinder

The cylinder is defined in the coordinate system γ (blue) of the image 04 (green) such that:

• cylinder axis is equal to c2 axis of γ,
• image plane of the image 04 is tangential to the cylinder,
• image coordinate system defined on the cylinder surface has square pixels and equal horizontal resolution in the line where it touches the image 04.

This leads that the cylinder has radius = 1 in the γ system. The cylinder surface is a set of points such that

For parametrisation of the cylinder surface we define two coordinate systems: the first (magenta) is based on γ, second (red) is pixel-based system of panoramic image.

First, the surface of the cylinder is parametrised by circumference length (equal to angle since radius=1) and by the coordinate. The ray intersects the cylinder in the point

Second, the pixel size must be applied and image origin adjusted. Since the horizontal pixel size is 1/K(1,1) measured in units given by γ system, this leads to

where the origin must be found such that all pixels from the input images will fit into the resulting panorama.

1. 06_data.mat
2. 06_histograms.pdf
3. 06_borders.pdf
4. 06_panorama.png
5. 06_borders_c.pdf
6. 06_panorama_c.png


text:
# The Calderón problem for connections

Cekić, Mihajlo (2017-10-03)

Thesis

This thesis is concerned with the inverse problem of determining a unitary connection $A$ on a Hermitian vector bundle $E$ of rank $m$ over a compact Riemannian manifold $(M, g)$ from the Dirichlet-to-Neumann (DN) map $\Lambda_A$ of the associated connection Laplacian $d_A^*d_A$. The connection is to be determined up to a unitary gauge equivalence equal to the identity at the boundary. In our first approach to the problem, we restrict our attention to conformally transversally anisotropic (cylindrical) manifolds $M \Subset \mathbb{R}\times M_0$. Our strategy can be described as follows: we construct the special Complex Geometric Optics solutions oscillating in the vertical direction, that concentrate near geodesics and use their density in an integral identity to reduce the problem to a suitable $X$-ray transform on $M_0$. The construction is based on our proof of existence of Gaussian Beams on $M_0$, which are a family of smooth approximate solutions to $d_A^*d_Au = 0$ depending on a parameter $\tau \in \mathbb{R}$, bounded in $L^2$ norm and concentrating in measure along geodesics when $\tau \to \infty$, whereas the small remainder (that makes the solution exact) can be shown to exist by using suitable Carleman estimates. In the case $m = 1$, we prove the recovery of the connection given the injectivity of the $X$-ray transform on $0$ and $1$-forms on $M_0$. For $m > 1$ and $M_0$ simple we reduce the problem to a certain two dimensional $\textit{new non-abelian ray transform}$. In our second approach, we assume that the connection $A$ is a $\textit{Yang-Mills connection}$ and no additional assumption on $M$. We construct a global gauge for $A$ (possibly singular at some points) that ties well with the DN map and in which the Yang-Mills equations become elliptic. By using the unique continuation property for elliptic systems and the fact that the singular set is suitably small, we are able to propagate the gauges globally. For the case $m = 1$ we are able to reconstruct the connection, whereas for $m > 1$ we are forced to make the technical assumption that $(M, g)$ is analytic in order to prove the recovery. Finally, in both approaches we are using the vital fact that is proved in this work: $\Lambda_A$ is a pseudodifferential operator of order $1$ acting on sections of $E|_{\partial M}$, whose full symbol determines the full Taylor expansion of $A$ at the boundary.


text:
## Introduction

Photodynamic therapy (PDT) has become a major strategy for solid cancer treatment due to its inherent advantages, such as high selectivity, noninvasiveness, and low systemic toxicity. Effective PDT typically involves the use of photosensitizers (PSs), appropriate laser irradiation, and a sufficient oxygen supply1. Specifically, the electrons of PS are elevated from the ground state (S0) to the excited state (S1) upon light irradiation and then pass through the excited triplet state (T1) through intersystem crossing, forming free radicals through a type I reaction or transferring energy to the surrounding molecular oxygen (3O2) to produce highly active singlet oxygen (1O2), as illustrated schematically in Fig. 1 (ref. 2). The generated 1O2 and adjacent biological macromolecules undergo oxidative reactions that can further lead to cytotoxicity, cell damage, and even death. While it has numerous advantages, the application of PDT for cancer treatment is limited due to several intrinsic drawbacks. For instance, when subjected to light irradiation, PS will convert oxygen to reactive oxygen species (ROS) with high cytotoxicity, which further aggravates the hypoxia of tumors. Therefore, in addition to pre-existing tumor hypoxia, the effectiveness of PDT will be further deteriorated by the oxygen consumption involved in the generation of ROS3. Moreover, the deterioration of tumor hypoxia will further promote cancer progression and metastasis, increasing the risk of PDT resistance4. Therefore, proposing new and effective strategies to reduce tumor hypoxia has become a primary research focus for improving PDT efficacy.

Recently, many studies on improving the hypoxic status of tumors to enhance PDT efficacy have been carried out, and significant progress has been made5,6,7,8. For instance, novel nonreactive oxygen carriers, such as liposome-encapsulated hemoglobin (HB) and perfluorocarbon carbide, have been designed and used9. Moreover, reactive oxygen supply materials, such as metal oxides, hydrogen peroxide (H2O2) enzymes, active nanoparticles that can react with acid and H2O2 in vivo to produce oxygen, and materials that can produce oxygen under PDT conditions, such as PT(IV) nanocomposites, have also been studied. In addition, other strategies that can alleviate or take advantage of hypoxic environments have also been reported, such as administering hypoxia-inducible factor inhibitors, targeting organelles, increasing blood perfusion, and performing combined therapy with hypoxia-response drugs.

Therefore, a comprehensive and in-depth depiction of the whole scene of the recent development of tumor hypoxia-improving strategies to enhance PDT efficacy, from fundamentals to applications, is desirable. In this review, we summarize the main research strategies in recent years, from the design of novel nonreactive oxygen carriers to reactive materials and other strategies, including the regulation of tumor microenvironments and PDT-involved multimodal therapy, to improve the hypoxic microenvironment to enhance PDT efficacy (Fig. 2).

## Nonreactive oxygen carriers

### Hemoglobin

HB is one of the major components in red blood cells with inherent reversible oxygen-binding capability and can transport oxygen molecules to tissues. However, as an oxygen donor, cell-free HB presents several problems, including poor stability, short circulation time, and potential side effects10. Therefore, the incorporation of HB into hybrid oxygen carriers, such as glutaraldehyde-polymerized HB and poly(ethylene glycol)-conjugated HB (PEG-HB), to form HB-based oxygen carriers (HBOCs) to facilitate PDT has been widely investigated11,12,13,14. However, most HBOCs cause vascular contraction due to their nitric oxide (NO) scavenging capacity, leading to an unfavorable sharp increase in blood pressure15. Compared with HBOCs, oxygen carriers with sizes in the nanometer range can perfuse tumor tissues more efficiently and provide a higher oxygen supply in hypoxic tumors. Many studies on the development of HB-based nano-oxygen carriers have been carried out, and significant progress has been achieved16,17.

Liposomes have attractive prospects in clinical applications due to their inherent advantages, such as good biocompatibility, targeting ability and drug protection. Recently, liposome-encapsulated HB has been proposed and studied as an oxygen carrier and was found to reduce side effects and enhance the half-life of the HB cycle. For example, as illustrated in Fig. 3, Yang et al.18 developed a multifunctional liposome with enhanced chemotherapeutic effects by simultaneously incorporating HB as the oxygen carrier and doxorubicin (DOX) as the antitumor drug. First, HB can be cross-bonded in the phospholipid membrane by hydrophobic forces. Then, DOX and HB are loaded into the aqueous phase of the liposome, while part of the HB is connected to the surface of the liposome membrane through hydrophobic interactions to form DOX–HB–liposomes (DHL). The prepared DHL can offer sufficiently high oxygen loading and can perform site-specific delivery of oxygen to tumors, which can improve tumor hypoxia. MTT results indicated that even at a concentration of 150 μg/mL, the cytotoxicity to the test cells was very low; hematoxylin and eosin (H&E) staining results showed that there was no obvious histological damage to normal organs, which further proved the safety and biocompatibility of liposomes.

Studies have shown that excessive iron can promote the occurrence and growth of tumors, and many types of cancer cells (such as breast cancer, liver cancer, and colon cancer) upregulate proteins related to iron uptake. Therefore, cancer cells may absorb iron-containing HB more easily, allowing such HB-liposomes to be specifically internalized into cancer cells. Consistently, the results of an in vivo distribution experiment with HBL revealed that compared with liposomes without HB, HBL showed stronger fluorescence at the tumor site, indicating that the HB-mediated drug combination could improve specific uptake into tumor cells. Moreover, under the effect of O2 interference, hypoxic cancer cells showed an increased capacity to take up the delivered drugs, resulting in significant cancer cell apoptosis. In addition, DHL can markedly increase ROS production in hypoxic tumors and thereby increase the cytotoxicity of DOX, which is typically mediated by ROS. Due to the resultant higher drug and oxygen concentrations in tumor regions, DHL shows great promise in reversing hypoxia-induced chemoresistance and exhibits improved antitumor effects.

In addition, Cai et al.19 developed biomimetic lipid-polymer nanoparticles (I-ARCs) that can load a special compound assembled from indocyanine green (ICG) as a PS and HB as an oxygen carrier (Fig. 4). The prepared I-ARCs can act as nanosized artificial red blood cells with the ability to generate oxygen and provide real-time monitoring of the PDT process simultaneously. Specifically, I-ARCs consist of biocompatible lipid shells, which were designed to mimic the cell membrane, and poly(d,l-lactic acid-co-glycolic acid) (PLGA) nuclei, in which HB was assembled with ICG to form HB/ICG complexes. Upon 808 nm light irradiation, I-ARCs can produce 9.5 times more ROS than deoxidized ICG nanoparticles. Moreover, since ICG shows fluorescence signals and a photoacoustic response, the biodistribution and metabolism of HB/ICG complex-loaded I-ARCs can be easily monitored during PDT.

### Perfluorocarbon

Considering that HB may be susceptible to conformational changes during the chemical modifications involved in the fabrication process of HB-based nanomaterials, other candidates, such as solutions or materials with considerable oxygen storage capacity, would be suitable alternatives. For example, compared with HB, perfluorocarbons (PFCs) can exhibit a higher oxygen solubility (40–56 ml O2 per 100 ml PFC at 760 Torr and 25 °C)20,21 and can donate molecular oxygen in the tumor tissue, thereby improving PDT efficiency in the hypoxic region. Recently, many efforts have been made to improve PDT efficacy by adding PFCs in preclinical or clinical research, as summarized in Table 122,23,24,25,26,27,28,29,30. For instance, Cheng et al.31 explored oxygen self-enrichment PDT (Oxy-PDT) by loading the PS IR780 into PFC nanodroplets to enhance the PDT effect (Fig. 5). In this system, a continuous oxygen supply facilitates the prolongation of PDT and fully utilizes the efficacy of IR780. The results demonstrated that compared with traditional PDT methods, the tumor growth of mice can be significantly inhibited by intravenously injecting a single dose of Oxy-PDT.

Although PFCs have a high oxygen capacity, the oxygen release efficiency is purely based on oxygen concentration gradient-dependent passive diffusion and is lower than that of HB. To overcome this obstacle, Song et al.32 developed an ultrasound-triggered tumor oxygenation strategy by using nano-PFC as the oxygen carrier, as shown by the schematics in Fig. 6. A human serum albumin-stabilized PFC nanodroplet system was proposed to regulate tumor-specific oxygen delivery and readily trigger oxygen release by external synchronous ultrasound therapy. The nanodroplets were intravenously injected into tumor-bearing 4T1 mice, and then the mice were treated with pure oxygen breathing for 30 min and ultrasonication for 30 min. In this process, nanodroplets adsorbed oxygen in the lungs, carried it into tumors through blood circulation, and then released oxygen effectively in the tumors. The level of O2 in the tumors increased rapidly from 17 to 49%, improving the antitumor effect of the PDT regimen.

### Oxygen-containing microbubbles/nanobubbles

In addition to HB and PFCs as oxygen carriers, oxygen-loaded microbubbles/nanobubbles have recently been used as a novel potential reagent to provide supplemental oxygen in the hypoxic microenvironment of solid tumors33,34. These bubbles have excellent biocompatibility and effective and high oxygen-carrying capacity (>70% v/v). The combination of microbubbles/nanobubbles and PSs has been demonstrated to have great potential to enhance therapeutic efficacy in hypoxic tumor models. In general, nanobubbles bound to PS can accumulate in the cytoplasm, and then the PS inside is released by destroying the shell of the bubbles. When the oxygen diffuses out from the shell, the bubble shrinks to a point where the external Laplace pressure breaks the bubble, thereby releasing the PS. Then, the PS generates singlet oxygen under laser irradiation and exerts an efficient PDT effect in a high oxygen environment.

For example, Song et al.34 designed bilaminar oxygen nanobubbles with chlorin e6 (Ce6) connected to a polymer shell as a new oxygen supply material. The developed oxygen-loaded nanobubbles can provide a large amount of oxygen to improve the therapeutic effect of oxygen-consuming PDT (Fig. 7). The lipid-polymer bilaminar oxygen nanobubbles were prepared using a combination of emulsion solvent evaporation and internal phase separation. Compared with the phospholipids or surfactants used in other oxygen delivery systems, the lipid molecules around the polymer shell selected by the researchers show good biocompatibility. The combination of the hydrophilic polymer PEG and lipid provides an invisible layer of nanocapsules, thereby inhibiting the phagocytosis of mononuclear macrophages and improving the stability of nanocapsules and system circulation life. More importantly, the hydrophobic fluorinated cap at the end of the polycaprolactone polymer provides a strong gas diffusion barrier to reduce the premature release of oxygen before reaching the tumor site. In addition, the polymer shell can withstand higher Laplace pressures, keeping the bubbles at the nanometer scale, thereby enabling the use of enhanced permeability and retention to target tumor cells while maintaining good stability. Moreover, the synthesized nanobubbles have good stability that can inhibit premature oxygen release and can be stored in the form of freeze-dried powders to enable simple pre-use reconstruction, thus solving the storage problems troubled by PFC oxygen delivery systems. Both in vitro and in vivo results showed that PDT treatment with oxygen-containing nanobubbles can greatly enhance the production of singlet oxygen and achieve excellent therapeutic effects.

## Reactive oxygen supplementation materials

### PDT-dependent materials

In recent years, researchers have prepared PDT-dependent materials for oxygen generation35,36, which can effectively supplement oxygen in PDT by decomposing nanoparticles under laser irradiation. Pt(IV) is a Pt-based light-activated precursor drug that can be decomposed into oxygen and Pt(II) with anticancer activity under long-wavelength light irradiation. The PS can therefore be assembled with Pt(IV) to form a nanocomplex for photodynamic-chemotherapy combined treatment. For example, Xu et al.37 designed self-generated multifunctional nanocomposites containing oxygen and Pt(II) to reverse the PDT resistance induced by hypoxia, as illustrated in Fig. 8. The nanocomposites consist of Pt(IV) and Ce6, by which upconversion nanoparticles (UCNP) are loaded to convert near-infrared (NIR) light at 980 nm to light emission at 365 and 660 nm, respectively, further inducing the decomposition of Pt(IV) and the production of oxygen and thereby properly driving PDT with UCNP-embedded nanoparticles (denoted UCPP). When laser irradiation at 980 nm is used to trigger the decomposition of UCPP, oxygen can be generated through chemical reactions to compensate for oxygen consumption in the PDT process, and active Pt(II) can be released for synergistic photochemistry.

### PDT-independent materials

In addition to PDT-dependent materials, PDT-independent materials could generate oxygen through chemical reactions not involved in the PDT process. Various reactive oxygen-generating materials, including metal oxides, catalase-active materials, and water-splitting materials, have been used to overcome hypoxia.

#### Metal oxide materials

In recent years, metal oxides have been demonstrated to produce oxygen through chemical reactions with acids or H2O2 and can effectively alleviate the oxygen consumption in the PDT process38,39,40,41. Therefore, metal oxides show promise for use along with PSs to improve the efficiency of PDT.

### MnO2 nanomaterials.

Manganese oxide (MnO2) has attracted increasing attention due to its ability to generate excessive oxygen with the aid of acid and H2O2 in the tumor microenvironment.

The following reactions illustrate how acid-induced H2O2 promotes the degradation of MnO2 and the resultant generation of oxygen:

$${\mathrm{MnO}}_2 + 2{\mathrm{H}}^ + \to {\mathrm{Mn}}^{2 + } + {\mathrm{H}}_2{\mathrm{O}} + 1/2{\mathrm{O}}_2;$$
(1)
$${\mathrm{MnO}}_2 + {\mathrm{H}}_2{\mathrm{O}}_2 + 2{\mathrm{H}}^ + \to {\mathrm{Mn}}^{2 + } + 2{\mathrm{H}}_2{\mathrm{O}} + {\mathrm{O}}_2;$$
(2)

MnO2 shows high reactivity and specificity to acidic environments and the presence of H2O2, where it can produce oxygen continuously to overcome hypoxia in tumors42,43,44,45. After reacting with excessive H2O2 in an acidic tumor microenvironment, MnO2 nanosheets can be degraded to produce a large amount of oxygen, thus greatly improving the efficiency of PDT. In addition to increasing the oxygen content, MnO2 can also be used as a glutathione (GSH) inhibitor in vivo to increase PDT efficacy. Moreover, it can be used to make upconversion nanocomposites to stimulate PSs at various wavelengths to enhance the selectivity of PDT. In addition, Mn2+ produced by MnO2 is a strong T1 magnetic resonance (MR) contrast agent for tumor detection, which can also be used for in site tumor imaging46,47. At the same time, it is worth mentioning that, as an essential trace element in the human body, Mn can be adjusted by metabolism, so it has good biocompatibility.

### MnO2 as a GSH inhibitor to replenish oxygen and increase PDT efficacy

In addition to low oxygen, the effectiveness of PDT is also limited by the hypoxia-induced overexpression of GSH, which consumes ROS before they reach their target and hence greatly reduces the efficiency of PDT. Numerous studies have attempted to develop a more stable and effective PDT system by the incorporation of GSH depletors48. MnO2 can be used as an efficient glutathione dehydrogenase inhibitor to reduce GSH levels; therefore, a MnO2-containing PDT system is expected to overcome the drawbacks caused by hypoxia and GSH overexpression at the tumor site49.

Sun et al.50 proposed a simple and convenient method for achieving the controllable growth of ultrathin manganese dioxide nanosheets by introducing a redox reaction on polydopamine nanospheres (Fig. 9). The prepared multifunctional polydopamine@ultrathin manganese dioxide/methylene blue (MB) nanoflowers (termed PDA@ut-MnO2/MB NFs) are suitable platforms for synergistic tumor therapy and can efficiently modulate the tumor microenvironment by generating O2 and depleting GSH. In addition, they can support ultrasensitive reduction-responsive MR imaging (MRI). The morphology and composition of PDA@ut-MnO2 were identified and characterized by scanning electron microscopy (SEM), transmission electron microscopy (TEM), high-resolution transmission electron microscopy (HRTEM) and high-angle annular dark field scanning transmission electron microscopy (HAADF-STEM), and their therapeutic effects were evaluated in mice bearing HCT116 colorectal tumors as models. Tumors were collected 15 days after injection, and PDA@MnO2/MB showed nearly complete tumor growth inhibition under MRI-guided PDT/PTT cotreatment. The H&E staining results of tumor sections also confirmed serious pathological damage, such as severe nuclear pyknosis and nuclear necrosis. Moreover, MRI of tumor-bearing mice found that the T1-weighted MR signal of the kidney was significantly enhanced 2 h after the injection, and the total manganese content in urine was measured by inductively coupled plasma mass spectrometry (ICP-MS). Approximately 65% of the manganese content was detected, suggesting that Mn2+ was eliminated from the body through the kidneys, and the in vivo toxicity was minimal. These results demonstrated that PDA@MnO2/MB NFs can achieve a high degree of tumor growth suppression under MRI-assisted treatment.

Furthermore, Fan’s group51 developed an intelligent PS-MnO2 nanosystem, which exhibited high intracellular drug delivery capacity and enhanced PDT efficacy by reducing GSH levels and increasing oxygen content in cancer cells. Specifically, MnO2 nanosheets act as oxygen supplements and oxidants to reduce intracellular GSH levels. In addition, they have been shown to effectively induce the fluorescence quenching of Ce6 to offer an on–off fluorescence signal for real-time monitoring. In the Ce6-MnO2 nanosystem, Ce6 can be well protected from light-induced self-destruction and can be effectively transported to the cytoplasm. After endocytosis, MnO2 nanosheets were reduced by intracellular GSH. The results show that the nanosystem can be disintegrated, generate oxygen, and reduce the GSH level, subsequently inducing 95% cell death when combined with Ce6. Moreover, the fluorescence recovery, accompanied by the dissolution of MnO2 nanosheets, can be used as a reporting signal for monitoring the transmission capacity.

### Upconversion nanocompounds loaded with MnO2 to replenish oxygen and increase PDT selectivity

Under the effect of upconversion nanocompounds loaded with MnO2, NIR light can be turned to ultraviolet (UV)/visible light with a broad emission spectrum; therefore, MnO2-loaded upconversion nanocompounds can also be applied as powerful tools to activate various PSs. Meanwhile, MnO2 can be decomposed in vivo to produce oxygen and further enhance PDT. As a result, NIR light-mediated upconversion nanocompounds, especially lanthanum-doped upconversion nanocrystals (UCNs) combined with MnO2, have been systematically investigated to effectively induce cell extirpation.

Xing’s group52 demonstrated the successful establishment of PS-loaded UCN nanoconjugates (PUNs) that can enhance the effectiveness of PDT by reducing the population of tumor-associated macrophages (TAMs) and hypoxia. PUNs can be prepared by combining MnO2 nanosheets with hyaluronic acid (HA) biopolymers (Fig. 10). When subjected to the acidic tumor microenvironment with excessive H2O2, MnO2 nanosheets can be degraded and produce a large amount of oxygen, which can greatly improve the efficiency of oxygen-dependent PDT. Moreover, by transitioning NIR light to UV/visible light, PUNs can also offer a high degree of tissue penetration and help solve the safety problems involved in conventional PDT techniques. In addition, due to the existence of biologically stimulated HA, PUNs can drastically decrease the tumor recurrence rate by reprogramming the propagation of tumor-promoting M2 TAMs into tumor-unfavorable M1 macrophages. This strategy provides an excellent opportunity for enhanced cell ablation in NIR-mediated PDT therapy by weakening the hypoxic tumor microenvironment and therefore inspires new strategies for immunotherapy that can prevent tumor recurrence in the later stage of PDT.

In addition, mesoporous silica nanospheres incorporating fine CaF2:Yb and Er upconversion nanocrystals were synthesized by the thermal decomposition method by Gu et al.53. Then, a new nano-PDT platform was obtained by coating the nanospheres with a thin layer of manganese dioxide and loading the PS Ce6. In the composite nanoparticles, Mn2+ ions doped in the CaF2 crystal lattice effectively enhanced the red upconversion luminescence triggered by NIR light, which stimulated the adsorption of Ce6 by resonance energy transfer, thus improving the photodynamic phenomena. At the same time, the MnO2 coating regulated the hypoxic microenvironment of tumors by the in situ generation of O2 through reacting with endogenous H2O2 of tumors. These two mechanisms play important roles in the treatment of NIR-triggered photodynamic tumors. After Ce6 loading, the nanoparticles showed good stability in physiological solutions (including ultrapure water, phosphate-buffered saline, fetal bovine serum, and cell culture medium (RPMI-1640) containing 10% serum) without any agglomeration. More importantly, the results of in vivo experiments showed that the tumor shrank significantly under NIR light irradiation and remained small for 14 days without recurrence.

UCNPs loaded with MnO2 based on this unique material design strategy not only have the anticancer prospects shown above but also offer guiding significance for the development of other types of functional nanostructures.

### Other metal oxide nanomaterials

Lan et al.54 developed novel nano metal–organic framework-based metal oxide nanoparticles by using 5,10,15,20-tetra(p-benzoic acid) porphyrin (TBP) ligands and Fe3O clusters to form Fe-TBP, as illustrated schematically in Fig. 11. O2 was generated by the Fenton-like reaction of Fe3O clusters and then converted into cytotoxic singlet oxygen (1O2) by photostimulated porphyrin. PDT mediated by Fe-TBP can trigger a systemic antitumor response to improve alpha-PD-L1 ICB (immune checkpoint blocker), resulting in the disappearance of microscopic effects in treating primary and untreated distant tumors. Treatment with Fe-TBP plus alpha-PD-L1 can induce the proliferation of CD4+ and CD8+ T cells, which infiltrate distant tumors to induce exfoliation. In a mouse model of colorectal cancer, Fe-TBP-mediated PDT significantly improved the therapeutic effect of alpha-PD-L1. A sustained abscopal effect was induced, both the primary tumor under the treatment and the untreated distant tumor regressed >90%. Moreover, ICP-MS analysis showed that Fe-TBP mostly gathered at the tumor site after 4 h of incubation and was cleared within 10 days, and no dark regions indicating toxicity of Fe-TBP were observed. This study proposes a new strategy combining enhanced PDT with immune therapy to induce systemic antitumor therapy.

Moreover, due to the spontaneous cycling of cerium dioxide (CeO2) between Ce3+ and Ce4+ in redox reactions, CeO2 can react with high levels of endogenous H2O2 in tumor cells and simultaneously generate H2O and O2. Therefore, CeO2 can be used as an optional O2-evolving agent to enhance the efficacy of PDT. Zeng et al.55 developed a core–shell Bi2S3@Ce6–CeO2 nanocomposite. Bi2S3 showed good photothermal, photoacoustic, and computed X-ray tomography (CT) imaging performance, and the PS Ce6, which has good photodynamic effects, was adsorbed on the surface of Bi2S3. Finally, CeO2, an O2-evolving agent, was coated on the Bi2S3@Ce6 surface to provide an in situ oxygen supply. The biocompatibility of Bi2S3@Ce6–CeO2 was evaluated by the MTT method, and the results showed that at concentrations of 12.5–200 μg/mL, 4T1 cells had a survival rate of more than 95% within 24 h, even at the maximum concentration of 200 μg/mL. The survival rate at 48 h also exceeded 90%. Similarly, Bi2S3@Ce6–CeO2 showed consistently low toxicity in mouse embryonic fibroblast NIH 3T3 cells. Moreover, the results of in vivo treatment showed that the tumor volume of the mice in the Bi2S3@Ce6–CeO2 group was close to zero, and the CD31 and Ki67 immunohistochemical results also showed significant damage to the microvessels of CD31-positive tumors and inhibited proliferation of positive cells. Multifunctional nanomaterials that combine photodynamic/photothermal/oxygen supply properties have shown improved tumor therapy efficacy both in vitro and in vivo.

#### Catalase-active material

Catalases capable of triggering the decomposition of H2O2 into water and oxygen are enriched in malignant tumor cells, with concentrations higher than those in healthy cells, and are closely related to the progression and proliferation of tumors. Such materials can also be used as an oxygen source to trigger PDT56,57,58,59. Based on this understanding, the concept that catalase-combined PS can enhance PDT efficiency by the synergistic promotion of in situ oxygen production in tumor tissues has been proposed and demonstrated.

For example, gold nanoclusters (AuNCs) have been widely used as materials with catalase activity in recent years60,61. Nanomaterial-based enzymes can be used as artificial catalases with controllable release to use catalase-like activities to catalyze the transformation of intratumoral H2O2 into O2 to alleviate tumor hypoxia. Dan et al.61 reported the use of ultrasmall AuNCs-ICG as nanozymes with theranostic features, efficient renal clearance, and superior catalase-like activity to modulate tumor hypoxia and enhance PDT and radioisotope tracer (RT) (Fig. 12). AuNCS effectively decompose a large amount of H2O2 in the tumor to produce oxygen, thus improving the naturally hypoxic tumor microenvironment and enhancing conventional PDT and RT efficacy.

Moreover, taking advantage of Pt’s catalase activity to change H2O2 into oxygen, Wei et al.62 designed novel two-dimensional PdS-based nanomaterials with good biocompatibility, good near-infrared absorption characteristics, and high tumor enrichment ability, which show great promise in the diagnosis and treatment of tumors. In particular, Pd@Pt-PEG-Ce6 nanocomposites were designed by using Pd–Pt two-dimensional palladium nanocomposites as carriers of PSs, as shown in Fig. 13. Pd@Pt-PEG-Ce6 can effectively deliver PS to tumor cells and decompose endogenous H2O2 to facilitate oxygen generation, thus significantly enhancing the efficacy of PDT. Moreover, the PDT efficacy of Pd@Pt-PEG-CE6 can also be improved by the mild photothermal effect of Pd:Pt nanoplates.

In addition, the combination of catalase and a PS to achieve a better PDT effect has been investigated. Hu’s group58 developed a new type of oxygen self-sufficient PDT platform by coloading with catalase and MB to construct zeolite-catalase-MB nanocapsules (ZCM nanocapsules), as illustrated by the SEM, TEM, and HRTEM images in Fig. 14. The prepared ZCM nanocapsules showed 90% relative activity compared to equivalent free catalase. Under real-time ultrasound imaging guidance, ZCM nanocapsules can be precisely implanted into the tumor area. Since they can decompose endogenous H2O2 in a sustained manner and provide in situ O2 generation, the ZCM nanocapsules can effectively improve hypoxia and enhance intratumoral ultrasound contrast. Moreover, the incorporated MB can prevent the rapid leaching of the PS and contribute to its sustained release of PS at tumor site. In vitro results demonstrated that local PC cells can be completely killed upon near-infrared laser irradiation, and no therapy-induced side effects or tumor recurrence were observed. As described above, the use of catalase-active materials in combination with PSs and the formation of new processing systems through reasonable design strategies will help overcome the challenges in PDT and improve its efficacy.

#### Water-splitting materials

Current strategies for oxygen supplementation typically involve the use of biological oxygen-producing materials such as MnO2 and catalase-like materials, but their O2 yield may be limited by intracellular restricted H2O2 concentrations. In nature, plants produce oxygen efficiently from abundant H2O through photosynthesis. Inspired by this phenomenon, water-splitting materials, such as nanocomposites based on inorganic, organic, macromolecular, or hybrid materials, have been developed to produce H2 and O263,64,65. As the most abundant compound in organisms, water can provide almost unlimited O2 production compared with other oxygen-producing materials. At present, the main water-splitting materials used to alleviate PDT hypoxia therapy are calcium peroxide (CaO2) nanoparticles, carbon nitride (C3N4) nanocomposites, and TiO2 nanotubes.

### Calcium peroxide (CaO2) nanoparticles

CaO2, a safe water-splitting material with high biocompatibility, rapid cell metabolism, and efficient O2 production, is considered to be a promising material to regulate solid tumor hypoxia and improve PDT efficiency66,67. It can generate a large amount of O2 upon coming into contact with water to improve PDT efficacy. Typical CaO2-based oxygen-producing biomaterials are mainly fabricated by combining CaO2 with hydrophobic polymers.

For instance, Liu et al.68 synthesized the liposome-based nanoparticle LipoMB/CaO2 with oxygen self-enrichment properties (Fig. 15). Under laser irradiation, the singlet oxygen produced by the PS MB can induce lipid peroxidation and destroy the liposome structure, further exposing CaO2, expanding the contact area between CaO2 and H2O and producing a large amount of oxygen to alleviate hypoxia. In vivo and in vitro experiments have demonstrated that LipoMB/CaO2 shows superior oxygen production capability with an enhanced PDT effect that can effectively alleviate hypoxia in tumors.

### Carbon nitride (C3N4) nanocomposites

Among water-decomposing materials, C3N4 has attracted much attention due to its relatively narrow band gap (2.7 eV) and strong reduction ability. Water splitting by C3N4 can be triggered under high-penetration red light (>600 nm), making it suitable for in vivo therapy. Moreover, since C3N4 is not doped with metal elements, it has high biocompatibility, good stability, and nontoxicity, and can be used in the biomedical field69,70,71,72,73.

For example, Zhang’s group74 designed an intelligent nanoreactor R-NCNP with a combined water-splitting effect and multimodal imaging function (Fig. 16). This nanocomposite has a mesoporous C3N4 layer coated on the core–shell structure of nitrogen-doped graphene quantum dot (N-GQD)@hollow mesoporous silica nanosphere and then bound P-PEG-RGD amphiphilic polymers to achieve targeted delivery, resulting in the nanoreactor designated R-NCNP. Upon 630 nm laser irradiation, the PSs are activated to produce singlet oxygen, while H2O molecules are transformed into O2 to increase the tumor oxygen content through C3N4-mediated water-splitting. In addition, N-GQDs endow R-NCNP with both PTT/PDT and imaging capabilities. Experimental results show that R-NCNP can enhance the efficacy of PDT by alleviating hypoxia and assist with multimodal imaging for a good combination of diagnosis and treatment. Moreover, the distribution in the body showed only a weak fluorescent signal in the liver, possibly caused by phagocytosis of the nanoparticles by mononuclear phagocytes. In addition, a moderate accumulation of fluorescence was detected in the kidney tissue, indicating that the nanoparticles were excreted and eliminated from the kidney.

### TiO2 nanotubes

TiO2 materials have attracted extensive attention in the field of water-splitting materials due to their adjustable band gap, excellent photostability, high catalytic activity, low toxicity, and low cost75,76. They can provide large internal and external surface areas to improve the light absorption intensity and facilitate further modification. They can be used in combination with carbon nanodots (CDots) for PDT treatment via water-splitting processes:

$$2{\mathrm{H}}_2{\mathrm{O}} \to {\mathrm{H}}_2{\mathrm{O}}_2 + {\mathrm{H}}_2;$$
(3)
$$2{\mathrm{H}}_2{\mathrm{O}}_2 \to 2{\mathrm{H}}_2{\mathrm{O}} + {\mathrm{O}}_2.$$
(4)

Yang et al.77 designed carbon nanodot-modified TiO2 nanotubes (CDots/TiO2 NTS) to release oxygen and reactive oxygen species simultaneously in an anoxic environment. Under laser irradiation, the CDots/TiO2 PS will produce a large amount of singlet oxygen through an upconversion process to induce apoptosis or necrosis and supply a large amount of oxygen through continuous water splitting to improve the efficacy of PDT. Experimental results in vivo and in vitro show that CDots/TiO2 NTS have good PDT effects, which makes them an excellent photosensitive material with great potential for application in the biomedical field.

## Other strategies involving regulating tumor microenvironments and multimodal therapy including PDT

### Improving blood perfusion in tumors

With the rapid growth of solid tumors, the formation of a neovascular network lags behind cell proliferation, and the construction of a new microvascular network is irregular. The abnormal vascular network has temporary closure or “blind ends”, which a decrease of tumor blood perfusion and oxygen partial pressure (pO2) in the microenvironment, making it difficult for the tumor to obtain an adequate O2 supply to meet its metabolic needs, resulting in tumor hypoxia and reducing the therapeutic effect of PDT78,79,80,81. Therefore, improving blood flow perfusion by improving vascular conditions in tumors is a good approach to increasing oxygen content. It is mainly achieved by improving tumor hemodynamics via photothermal therapy (PTT) and regulating tumor microvessel structure using chemotherapeutic drugs.

#### Improving tumor hemodynamics

Tumor hypoxia is largely related to disordered and chaotic tumor blood flow, so improving blood flow has become an effective way to increase tumor O2 concentration. Many studies have shown that increasing local temperature by mild heating can effectively increase tumor blood flow and increase oxygen content in solid tumors82,83. Therefore, the combination of PDT and PTT is often used to treat cancer cells, as PTT can not only convert light energy into heat energy under the irradiation of an external light source to kill cancer cells but also be administered as a mild photothermal pretreatment to improve blood flow in solid tumors.

Our group84 designed a dual oxygenated nanoparticle MnO2@chitosan-CyI (MCC) for enhanced phototherapy, in which the PS CyI is a near-infrared dye with both PDT and PTT effects. When exposed to near-infrared radiation, the increase in tissue temperature caused by the photothermal effect accelerates blood flow and effectively alleviates tumor hypoxia. Moreover, MnO2 can not only reduce the level of GSH but also act as an efficient in situ oxygen generator, thus further increasing the production of ROS. This MCC nanosystem shows great promise as an intelligent and versatile nanotherapy platform that can improve the tumor microenvironment under hypoxia to obtain better therapeutic effects of PDT/PTT.

#### Regulation of tumor microvessel structure

In addition to using photothermal effects to increase blood flow velocity within solid tumors, chemotherapeutic drugs have recently been used to regulate the disordered microvascular structure of tumors (e.g., gemcitabine, cyclophosphamide, and cisplatin) to restore the physiological perfusion and oxygenation of tumor vessels more efficiently85,86,87. For example, Shen et al.88 found that thalidomide (Thal) can correct the imbalance between proangiogenic factors and antiangiogenic factors, resulting in decreased leakage of tumor vessels and increased thickness of blood vessels as well as tumor perfusion (Fig. 17). Finally, the tumor vascular system was normalized and tumor oxygenation was increased. Thal, as a new candidate drug, can maximize the efficacy of PDT for solid tumors if combined with a PS.

### Targeting mitochondria in tumors

A lack of sustained alleviation of hypoxia in solid tumors is one of the major problems for in situ oxygen-producing nanomaterials, since the uninterrupted high respiration in tumors consumes oxygen through mitochondria. As an indispensable organelle responsible for cell respiration, mitochondria may aggravate tumor hypoxia and limit the efficacy of PDT89,90. Hence, it is feasible to design nanocomposites that can selectively inhibit mitochondrial activity and produce O2 sustainably in hypoxic tumors.

For example, Yang et al.91 constructed photodynamic Mn3O4@MSNs@IR780 nanoparticles with sustainable hypoxia mitigation capabilities (Fig. 18). The alleviation of hypoxia is achieved through intratumor H2O2 catalysis and targeting mitochondria to inhibit respiration. These Mn3O4 nanoparticles responded to the H2O2-enriched tumor microenvironment by decomposing and catalyzing H2O2 to produce O2. Subsequently, IR780 was released and activated and spontaneously targeted mitochondria due to its natural mitochondrial affinity. Under laser irradiation, Mn3O4@esoporous silica nanomaterials (MSNs)@IR780 can destroy mitochondria and inhibit cell respiration, which can alleviate the persistent hypoxia of tumor tissue, thus improving the therapeutic effect of PDT. In vitro experiments showed that Mn3O4@MSNs@IR780 can continuously alleviate tumor hypoxia and provide a new way to overcome sustainable hypoxia.

### Downregulating HIF-1 expression in tumors

The high expression of hypoxia-inducible factor (HIF-1α) is one of the key barriers to PDT treatment. Although current in situ oxygen supply strategies can alleviate tumor hypoxia to some extent, the nonsustainable supply of O2 offered by these strategies can induce only moderate HIF-1α degradation, and the residual HIF-1α will still adversely affect the PDT effect. Using effective HIF-1α inhibition in hypoxic tumors is a feasible way to achieve adequate tumor oxygenation to enhance PDT efficacy. Inspired by HIF-1 intracellular signaling pathways, researchers have been working on the precise blocking of HIF-1α by small molecular inhibitors in recent years92,93.

Liu et al.94 utilized rapamycin (RAP) as an upstream activator of HIF-1 and mammalian rapamycin targeting gene (mTOR) inhibitor in cancer cells, whose inhibition of mTOR leads to HIF-1α expression and HIF-1 transcription-dependent blockade. As illustrated in Fig. 19, an efficient core–shell nanosystem was designed to alleviate hypoxia and improve PDT efficacy through simultaneous oxygenation and HIF-1α inhibition. The PSs Ce6 and RAP formed dual-drug nanopores through self-assembly and were then loaded with catalase. Catalase, as an oxygen provider, catalyzes H2O2 decomposition into O2 and leads to the sustained release of RAP to downregulate HIF-1α expression, thus achieving in situ oxygen generation and HIF-1α inhibition in tumor tissues, together relieving PDT pressure on hypoxia.

### PDT-activated hypoxia-amplified bioreductive therapy

In recent years, the application of hypoxia-reactive anticancer drugs combined with PDT under hypoxic conditions has aroused great interest. Unlike other strategies focused on alleviating hypoxia in tumors, this new approach takes advantage of the lack of oxygen in tumors. In addition, PDT-induced oxygen consumption can further enhance the role of hypoxia-activated drugs. Recent studies have demonstrated the feasibility and effectiveness of this strategy, in which the combined delivery of PS and prodrugs is achieved by nanoparticles, such as silica-shell nanoparticles or liposomes95,96,97,98.

Hypoxic response drugs usually produce highly cytotoxic substances under hypoxic conditions, and their cytotoxic activity can be triggered when combined with hypoxic induction therapy. The current combination of PDT with hypoxia-responsive drugs mainly involves telapamide (TPZ), TH-302, PR-104A, and benzoanthraquinone (AQ4N), which produce sufficient hypoxia in tumors by cotransporting with PS molecules and show strong cytotoxicity in the hypoxic microenvironment. Although this approach is rarely used in the field of PDT, its anticancer potential is certainly worth further exploration and confirmation.

Liu et al.99 designed double-layer silica-shell UCNPs, which can transfer the PS and bioreductive precursor drug TPZ together to achieve synergistic cancer treatment. UC-PDT was used to induce the cytotoxicity of active TPZ and produced a large amount of ROS. Compared with UC-PDT, TPZ UC/PS can significantly inhibit the growth of tumors, suggesting that TPZ exhibits a significant increase in cytotoxicity during PDT-induced hypoxia and has a profound impact on the therapeutic effect. This NIR-induced intratumoral hypoxia is sufficient to enhance hypoxia-induced bioreduction therapy, demonstrating an efficient synergistic approach to cancer treatment.

## Conclusions and outlooks

Because of its minimal invasiveness, high feasibility, and enhanced efficiency, PDT has made tremendous progress in anticancer therapy. The latest advances in oxygen supplementation nanoparticles have opened up promising perspectives to develop new PDT systems and provide multiple opportunities to circumvent the current PDT paradigm. On the one hand, exogenous molecular oxygen can be captured by biological, bionic, or physical mechanisms and then transported to solid tumors with the help of nanocarriers. On the other hand, through PDT-dependent materials, PDT-in-dependent materials or other methods, oxygen can be generated in vivo to provide a continuous local oxygen supply to improve PDT efficiency. Both strategies have demonstrated their potential in improving the oxygen content of PDT.

In this review, we present strategies to improve PDT efficiency by focusing on hypoxia, a key challenge in PDT-mediated cancer therapy. However, it is worth noting that although a large number of studies investigated improving PDT efficiency by relieving hypoxia in the tumor microenvironment, increasing the singlet oxygen yield of the PS itself is also an important means to enhance the efficiency of PDT. For example, we found that the heavy metal atom iodine can be modified in ICG PSs by the heavy atom effect and can significantly improve the singlet oxygen yield100. If such PSs are combined with the oxygen supplements described above, better PDT effects are expected. In addition, for clinical applications, PDT needs to meet many specific requirements, including biocompatibility and biodegradability. For example, when manganese dioxide nanosheets are used to produce oxygen, their degradation needs to be carefully evaluated in vivo. With these challenges seriously considered and addressed, overcoming tumor hypoxia in the use of PDT will become a powerful and versatile strategy for complicated cancer therapy.


text:
```> Hunh?  Like what kind of class is that?

The kind we were discussing before we wandered off topic and Frank
changed the Subject to ``Extended include'' at which point it caught

> You can't just go latex section1.tex if it doesn't have a
> \documentclass in it.  If it does, either you weren't using \input or
> \include to include it in the first place,

A proposed `journal' or `proceedings' class which redefines
\documentclass and other preamble commands such that `self standing'
articles *can* be input.

> A typical 300+-page document (source2e for example) doesn't tax TeX at
> all, and we are about to enter the era of web2c-7 with dynamic memory
> allocation.

Number of pages is more or less irrelevant to memory use. Source2e has
almost no cross references (sadly:-) so is quite light on memory
consumption. It is *slow* because all the verbatim bits are parsed `by
hand' to index command usage, and large parts of the old ASCII
latex209 comments are similarly parsed by hand so that commands
automatically print themselves verbatim even though they are not
marked up as LaTeX.

> add reason 2A) for using \include: To produce a partial document
> that is identical to part of the whole document.

Yes that as well, but to be reliable for that use the pages generated
have to be *exactly* correct and it is always difficult to be sure of
that with an include system. So you \includeonly{chapter3} but the