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Q: Distinguishing Two Compactifications of $[0,1)$ Pictured below are two subsets of the plane, each a compactification of the closed half-line with remainder a closed arc.  I am really frustrated by my inability to prove that the space pictured on the right, an arc with a ray seesawing back and forth toward it, is not homeomorphic to the closed topologist's sine curve.  How does one show this?
As well, in Nadler he refers to this object as the 'M-Continuum'.  I have seen the left figure also referred to as the M-Continuum by another author, where there are 'half-length dips' between each dip of the typical sine curve.  Are these two spaces homeomorphic?
I really have trouble figuring out how to compare these spaces.  Can anyone help?

Here is a better image of this other 'M-Continuum': http://hyperspacewiki.org/index.php/M_continuum

A: The space $X$ pictured on the right is homeomorphic to the closed topologist's sine curve $S$.
Write $X = L \cup R$ with $L = \{ 0 \} \times [-1,1]$ and $R$ the ray approaching from both sides to $L$. Parametrize $R$ by $u : (0,1] \to \mathbb{R}^2$. Define $L'= L \times \{ 0 \} \subset  \mathbb{R}^3$, $u' : (0,1] \to \mathbb{R}^3, u'(x) = (u(x),x)$ and $R'= u'((0,1])$. Then $X' = L' \cup R'$ is compact and the projection $p_{12} : \mathbb{R}^3 \to \mathbb{R}^2$ which forgets the third coordinate maps $X'$ homeomorphically onto $X$. The projection  $p_{23} : \mathbb{R}^3 \to \mathbb{R}^2$ which forgets the first coordinate maps $X'$ homeomorphically onto $X'' = p_{23}(X')$. But $X''$ is easily seen to be homeomorphic to $S$.
The M-continuum $M$ is not homeomorphic to $S$.
Let $M = L \cup R_M$ with a ray $R_M$ touching the $x$-axis at the bottom of each half-length dip and let $S = L \cup R_S$ with a ray $R_S$. Both $R_M, R_S$ touch the lines $y = \pm1$ at their big amplitudes. We call $r \in R_S$ resp. $r \in R_M$ an extremal point of type $\pm1$  if it lies on the line $y = \pm1$. The points $r \in R_M$ at the bottom of the half-length dips are called extremal point of type $0$.
We shall need
Lemma. Each homeomorphism $g : L \to L$ extends to a homeomorphism $G : S \to S$.
Proof. Write $g(0,y) = (0,g'(y))$ with a homeomorphism $g' : [-1,1] \to [-1,1]$. Each closed segment $\sigma$ of $R_S$ between two subsequent extremal points is a homeomorphic copy of $[-1,1]$. In fact, let $p_i : \mathbb{R}^2 \to \mathbb{R}$ denote the projection onto the $i$-th coordinate. Then $p_1$ restricts to a homeomorphism $p_1^\sigma : \sigma \to J^\sigma = p_1(\sigma)$ and  $p_2$ restricts to a homeomorphism $p_2^\sigma : \sigma \to [-1,1]$. $J^\sigma $ is the closed interval between the first coordinates of the given extremal points. We therefore get a homeomorphim $G^\sigma = (p_2^\sigma)^{-1} \circ g' \circ p_2^\sigma : \sigma \to \sigma$. The coordinate functions $G^\sigma_1 = p^\sigma_1 \circ G^\sigma : \sigma \to J_\sigma$ and $G^\sigma_2 = p^\sigma_2 \circ G^\sigma : \sigma \to [-1,1]$ are again homeomorphims. Note that $G^\sigma_2(x,y) = g'(y)$. The $G_\sigma$ and $g$ define a map $G : S \to S$ which is obvíously a bijection. It has the form $G(x,y) = (G_1(x,y),g'(y))$ where $G_1 : S \to [0,1]$ is obviously continuous (in fact, it maps $R_S$ homeomorphically onto $(0,1]$). This shows that $G$ is a homeomorphism.
Let us now assume that there exists a homeomorphism $h : M \to S$. We have $h(L) = L$ and $h(R_M) = R_S$. $g = h^{-1} : L  \to L$ extends to a homeomorphism $G : S \to S$. Then $H  = G \circ h : M \to S$ is a homeomorphism such that $H \mid_L = id$. Let $H_2(x,y)$ denotes the second coordinate of $H(x,y)$. By uniform continuity of $H$ we find $\delta > 0$ such that for all $y$ and $x <  \delta$
$$(\ast) \phantom{x} \lvert y - H_2(x,y) \rvert \le \lvert (0,y) - H(x,y) \rvert = \lvert H(0,y) - H(x,y) \rvert < \frac{1}{2}$$
Now consider three subsequent extremal points $e_1, e_2, e_3$ of $R_M$ of type $+1,0,+1$ with first coordinates $x_1 < x_2 < x_3 < \delta$. Let $e_i' = H(e_i) = (x'_i,y'_i)$. By $(\ast)$ we have $y'_1, y'_3 \in (\frac{1}{2},1]$, $y'_2 \in (-\frac{1}{2},\frac{1}{2})$. Let $\tau$ denote the closed segment of $R_M$ between $e_1$ and $e_3$. Then $H(\tau)$ is the closed segment of $R_S$ between $e'_1$ and $e'_3$. $H(\tau)$ begins and ends in the strip $\Sigma = (0,1] \times (\frac{1}{2},1]$ but is not contained in it ($e'_2 \notin \Sigma$). This is only possible if $H(\tau)$ passes through an extremal point of type $-1$. Let $(x_0,y_0) \in\tau$ such that $H_2(x_0,y_0) = -1$. We have $x_0 < \delta$ and $y_0 \in [0,1]$. Therefore $H_2(x_0,y_0) = -1$ must have a distance $<  \frac{1}{2}$ from $y_0$ which is impossible.
General methods to compare "arc-ray-spaces" with the closed topologist's sine curve $S$:
Start with the arc $L \subset \mathbb{R}^2$ and a ray $R \subset  \mathbb{R}^2$ such that $L \cap R =\emptyset$ and $Y = L \cup R$ is compact. The ray is given by an embedding $u = (u_1,u_2) : (0,1] \to \mathbb{R}^2$.
Compactness of $Y$ is equivalent to "$R$ approaches $L$" in the sense that for each open neigborhood $W$ of $L$ the remainder $R \backslash W$ is compact, in other words that there exists $x \in (0,1]$ such that $u((0,x)) \subset W$.
First assume that there exists a strictly decreasing sequence $x_n \in (0,1]$ which converges to $0$ such that
(1) $u_2$ is strictly increasing on all intervals $[x_{2k+1}, x_{2k}]$ and strictly decreasing on all intervals $[x_{2k+2}, x_{2k+1}]$.
(2) $u_2(x_{2k}) \to 1$ and $u_2(x_{2k+1}) \to -1$ as $k \to \infty$.
Then $Y$ is homeomorphic to $S$.
The first step is to "normalize" $Y$ as shown above, that is to replace $Y$ by a homeomorphic $Y' = L \cup R'$ such that $R'$ is parametrized by $u' = (u'_1,u'_2)$ with $u'_1(t) = t$. The sequence $(t_n)$ is replaced by a sequence $(t'_n)$ with properties (1) and (2) for $u'_2$.
We can find a self-homeomorphism $h$ of $(0,1]$ such that $h(\nu_n) = x'_n$ where $\nu_n = \frac{2}{(2n +1) \pi}$. Then we can replace $u'$ by $u' h$ without changing $R'$.Thus  w.l.o.g. assume that $x'_n = \nu_n$ (this will destroy $u'_1(x) = x$, but it is irrelevant).
Note that (1) implies $u_2(x_{n+1}) \ne u_2(x_n)$ for all $n$, and we let $J_n$ denote the closed interval bounded by these two points and $\phi_n : J_n \to [-1,1]$ the unique linear homeomorphism.
On each interval $I_n = [\nu_{n+1},\nu_n]$ there exists a homeomorphism $h_n$ such that $H_n = h_n \times \phi_n$ maps the segment of $R'$ lying over $I_n$  homeomorphically onto the corresponding segment of the closed topologist's curve. The $H_n$ and $id_L$ can be pasted to homeomorphism $Y' \to S$.
Next assume that there exist
(a) numbers $a > -1 $ and $b > 0$
(b) a decreasing sequence $x_n \in (0,1]$ which converges to $0$
such that
(1) $m_n = \min \{ u_2(x) \mid x \in [x_{2n},x_{2n-1}] \} \ge a$
(2) $M_n = \min(u_2(x_{2n}),u_2(x_{2n-1})) \ge m_n + b$.
Note that this requires $x_{2n} < x_{2n-1}$, but allows $x_{2n+1} = x_{2n}$.
Then $Y$ is not homeomorphic to $S$. This can be proved exactly as in the case of $M$.
If $Y$ and $S$ were homeomorphic, we would find a homeomorphism $H : Y \to S$ such that $H \mid_L = id$. We find $\delta > 0$ such that for all $y$ and $x <  \delta$
$$(\ast) \phantom{x} \lvert y - H_2(x,y) \rvert < \min(a+1,\frac{b}{2}) .$$
Now choose $n$ such that $x_{2n-1} < \delta$. Let $e_1 = u(x_{2n}), e_2 = u(x_{2n-1})$ and $e_i' = H(e_i) = (x'_i,y'_i)$. By $(\ast)$ we have $y'_i > M_n - \frac{b}{2}$ because the $y$-coordinate of $e_i$ is $\ge M_n$. Let $\tau$ denote the closed segment of $R$ between $e_1$ and $e_2$. Then $H(\tau)$ is the closed segment of $R_S$ between $e'_1$ and $e'_2$. $H(\tau)$ begins and ends in the strip $(0,1] \times (M_n - \frac{b}{2},1]$ but is not contained in it (there exists $x \in [x_{2n},x_{2n-1}]$ such that $M_n \ge u_2(x) + b$, i.e. $u_2(x) \le M_n - b$ and $H_2(u(x)) < M_n - \frac{b}{2}$). This is only possible if $H(\tau)$ passes through an extremal point of type $-1$. Let $(x_0,y_0) \in\tau$ such that $H_2(x_0,y_0) = -1$. We have $x_0 < \delta$ and $y_0 \ge m_n \ge a$. Therefore $H_2(x_0,y_0) = -1$ must have a distance $< a+1$ from $y_0 \ge a$ which is impossible.

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Botswana's premier wilderness and wildlife area, 1 of the 7 natural wonders of Africa
The Okavango Delta is a must-visit natural sanctuary in northwest Botswana, renowned for being the world's largest inland delta. Interestingly enough, it's also one of few that flow out onto land, not water. This UNESCO World Heritage site is a paradise of open grass plains, winding lagoons, mopane woodlands and extensive marshlands. An abundance of wildlife can be seen on both a permanent and seasonal basis thanks to attentive wildlife management and rich ecosystems maintained with the help of animals like the incredible hippos and elephants. This diverse area is also home to over 500 species of birds, around 1500 species of plants, and multiple endangered species like rhinos and wild dogs.
Seek out giraffes, zebras, and many species of antelope on a horseback safari.
Capture wildlife hunting their prey in wide-open spaces on a photography safari.
Get acquainted with the indigenous flora and fauna on a guided walking tour.
Encounter wildlife on a kayak safari through the delta with an expert guide.
Take a relaxing mokoro ride and explore the delta on these iconic dugout canoes.
Visit a traditional village to get insight into African culture and lifestyle.
Best time to go to Okavango Delta
A visit to the Okavango Delta is very rewarding year-round, but varying seasons present different highlights. Between June and October, the rains have come to an end and the water has reached the Delta, which is great for mokoro trips. This is the best time for wildlife viewing and you can expect lots of warm, sunny days with cooler nights. The rainy summer season from November to March brings hot days and afternoon rainstorms that are ideal for birding, appreciating newborn young and viewing predator activity.
Featured Okavango Delta stays
Duke's Camp
A striking safari destination on a vast wildlife-rich concession in the Okavango
From$880 /ppn
Explore stays in Okavango Delta

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text:
#include "TextRenderingPipeline.h"

#include <string>

#include <gloperate/gloperate.h>

#include <gloperate-text/Alignment.h>
#include <gloperate-text/LineAnchor.h>
#include <gloperate-text/stages/FontImporterStage.h>
#include <gloperate-text/stages/GlyphPreparationStage.h>
#include <gloperate-text/stages/GlyphRenderStage.h>

//#include "TextGenerationAlgorithm.h"
//#include "TextGenerationStage.h"
//#include "TextPreparationStage.h"
#include "GlyphSequenceDemoStage.h"

namespace
{
    const auto lorem =
R"(Lorem ipsum dolor sit amet, consetetur sadipscing elitr, sed diam nonumy eirmod tempor invidunt ut labore et dolore magna aliquyam erat, sed diam voluptua. At vero eos et accusam et justo duo dolores et ea rebum. Stet clita kasd gubergren, no sea takimata sanctus est Lorem ipsum dolor sit amet. Lorem ipsum dolor sit amet, consetetur sadipscing elitr, sed diam nonumy eirmod tempor invidunt ut labore et dolore magna aliquyam erat, sed diam voluptua. At vero eos et accusam et justo duo dolores et ea rebum. Stet clita kasd gubergren, no sea takimata sanctus est Lorem ipsum dolor sit amet.)";
}

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, wordWrap(true)
, lineWidth(400.f)
, alignment(gloperate_text::Alignment::Centered)
, lineAnchor(gloperate_text::LineAnchor::Baseline)
, optimized(true)
, targetFBO(nullptr)
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{
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    auto glyphPreparation = new gloperate_text::GlyphPreparationStage;
    auto glyphRendering = new gloperate_text::GlyphRenderStage;

    fontImport->resourceManager = resourceManager;
    fontImport->fontFilePath = fontFilename;

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    demo->pixelPerInch = pixelPerInch;
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    demo->origin = origin;
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    demo->lineAnchor = lineAnchor;

    glyphPreparation->font = fontImport->font;
    glyphPreparation->sequences = demo->sequences;
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        fontImport
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    ,   glyphPreparation
    ,   glyphRendering);
}

TextRenderingPipeline::~TextRenderingPipeline()
{
}

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text:
Back to graduate profiles
ESR13 – Joey Mark Diaz – based in Leeds
Project Title: Genomic Copy Number Alterations in Melanoma and Patient Survival
PhD Award Expected: Autumn 2020
In 2016, I joined the Leeds Institute of Cancer and Pathology (now Leeds Institute of Medical Research at St. James's), School of Medicine in the University of Leeds for a PhD study on analysis of genomic copy number alteration and melanoma survival as one of the selected 17 Marie Skłodowska-Curie research fellows under MELGEN.
I finished BSc in Statistics in Central Luzon State University, Philippines in 2010 as a government scholar with an undergraduate work on multiple imputation using Markov Chain Monte Carlo (MCMC) method. I then worked as research assistant in the National Genetic Program Unit of the Philippine Carabao Center in 2010 and as biostatistician in the University of Sto Tomas Center for Drug Research, Evaluation and Studies, Inc in 2011. I was awarded an Accelerated Science and Technology Human Resource Development Program (ASTHRDP) in 2011 by the Philippine Department of Science and Technology (DOST) to study MSc in Health Informatics-Bioinformatics in the University of the Philippines Manila. My graduate work focused on the classification of lung cancer using statistical and machine learning methods applied to transcriptomic data.
Aside from science, I have more than three years of corporate experience working on statistical financial risk models as credit scoring specialist in the RCBC Savings Bank and as retail modelling analytics manager in the Australia and New Zealand Banking Group Limited.
I look forward to taking part in improving patient treatment in my home country through personalised and stratified medicine using genomics.
http://melgen.org/wp-content/uploads/2018/07/01_Joey_Diaz_1.mp4
Váraljai R., Wistuba-Hamprecht K., Seremet T., Diaz JMS, Nsengimana J., Sucker A., Griewank K., Placke J-M., Horn P.A., von Neuhoff N., Shannan B., Chauvistré H., Vogel F.C.E., Horn S., Becker J.C., Newton-Bishop J., Stang A., Neyns B., Weide B., Schadendorf D., Roesch A. Application of Circulating Cell-Free Tumor DNA Profiles for Therapeutic Monitoring and Outcome Prediction in Genetically Heterogeneous Metastatic Melanoma. JCO Precision Oncology 2019; :3, 1-10. Available at: https://ascopubs.org/doi/10.1200/PO.18.00229
Poźniak J, Nsengimana J, Laye JP, O'Shea SJ, Diaz JMS, Droop AP, Filia A, Harland M, Davies JR, Mell T, Randerson-Moor JA, Muralidhar S, Hogan SA, Freiberger SN, Levesque MP, Cook GP, Bishop DT, Newton-Bishop J. Genetic and Environmental Determinants of Immune Response to Cutaneous Melanoma. Cancer Res. 2019 May 15;79(10):2684-2696. Available at: https://cancerres.aacrjournals.org/content/79/10/2684
Filia A, Droop A, Harland M, Thygesen H, Randerson-Moor J, Snowden H, Taylor C, Diaz JMS, Pozniak J, Nsengimana J, Laye J, Newton-Bishop JA, Bishop DT. High-Resolution Copy Number Patterns From Clinically Relevant FFPE Material. Sci Rep. 2019 Jun 20;9(1):8908. Available at: https://www.nature.com/articles/s41598-019-45210-2
Thakur R, Laye JP, Lauss M, Diaz JMS, O'Shea SJ, Poźniak J, Filia A, Harland M, Gascoyne J, Randerson-Moor JA, Chan M, Mell T, Jönsson G, Bishop DT, Newton-Bishop J, Barrett JH. Transcriptomic Analysis Reveals Prognostic Molecular Signatures of Stage I Melanoma. Clin Cancer Res. 2019 Dec 15;25(24):7424-7435. Available at: https://clincancerres.aacrjournals.org/content/25/24/7424
Muralidhar S, Filia A, Nsengimana J, Poźniak J, O'Shea SJ, Diaz JM, Harland M, Randerson-Moor JA, Reichrath J, Laye JP, van der Weyden L, Adams DJ, Bishop DT, Newton-Bishop J. Vitamin D-VDR signaling inhibits Wnt/beta-catenin-mediated melanoma progression and promotes anti-tumor immunity. Cancer Res. 2019 Dec 1;79(23):5986-5998. Available at: https://cancerres.aacrjournals.org/content/79/23/5986.


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Think disasters are isolated? Think again, warns the UN – Al Jazeera
Posted by Editor on Sunday, September 12th 2021
A new report makes the case that disasters are interconnected and that piecemeal solutions will continue to fall short.
Unless the international community changes its approach to not only ask 'what' happened when investigating disasters but to also also 'why', any preparatory measures will fall short, a new United Nations report warns [File: Hassan Ammar/AP Photo]
By Radmilla Suleymanova 8 Sep 2021
Given the ever-increasing frequency of severe weather events, human-made catastrophes and epidemics, piecemeal and fragmented responses will fail to address root causes and may in fact compound the challenges, a new United Nations report argues.
The Interconnected Disaster Risks report analyses 10 disasters of 2020 and 2021, including the Amazon wildfires, the Beirut explosion, and the cold wave in Texas in the United States among others, and makes the case that solving such problems will require addressing their root causes rather than surface challenges.
"If we keep trying to manage disasters as isolated events, we will fail," Jack O'Connor, senior scientist at the United Nations University's Institute for Environment and Human Security, told Al Jazeera.
"Unless we change our approach to not only ask 'what' happened when investigating disasters, but also 'why' they happened, any preparatory measures we devise will not be enough," said O'Connor, who is the lead author of the report.
To illustrate how disasters are far from localised, the report notes that the Amazon's human-made wildfires led directly to 2,195 people being hospitalised in South America. But in total, 4.5 million people worldwide were affected by the harmful levels of air pollution.
Disasters can also compound each other.
Last August in Beirut, Lebanon, 2,750 tonnes of ammonium nitrate detonated, causing 200 deaths, 6,000 injuries and up to $4.6bn in damage to local infrastructure.
That disaster further exacerbated the coronavirus pandemic outbreak in the country as already overwhelmed hospitals were hit by a wave of injured people.
And the COVID-19 pandemic in the border region of India and Bangladesh, where almost 50 percent of the population lives under the poverty line, made it extremely difficult to prepare for Cyclone Amphan.
Global disasters – whether in Texas or Beirut – should no longer be viewed in isolation, the UN University report says.
Houses, businesses and beaches are seen damaged at the entrance to Grand Isle, Louisiana in the United States after the town and barrier island was left devastated by Hurricane Ida [File: Adrees Latif/Reuters]
Disasters are also interconnected, the report argues, using the example of the link between the Arctic heatwave and the Texas cold wave.
In 2020, the Arctic experienced the second-highest air temperatures on record – a development that could lead to extreme cold spells and heatwaves in Europe and North America.
The February 2021 Texas cold wave left some four million people without electricity in a US state ill-prepared for freezing temperatures.
If kept at current levels, greenhouse gas emissions will continue to warm the Arctic, and climate shocks will become more frequent.
The coronavirus pandemic has illustrated just how hyper-connected society truly is, and has proven that no borders can contain disasters.
Other disasters the report examines include Central Vietnam floods; Chinese paddlefish extinction; the global COVID-19 pandemic; the Arctic heatwave; Cyclone Amphan; the Great Barrier Reef bleaching in Australia, and the desert locust outbreak in East Africa, the Middle East and the Indian subcontinent.
The report underscores the need to reduce human greenhouse gas emissions, as it found they were the root cause behind seven of the 10 events the UN University examined. Slowing down climate change to protect biodiversity and give ecosystems a chance to recover from the damage done is also critical.
For instance, a paddlefish fish going extinct in China is part of a larger, global freshwater fish extinction, O'Connor said.
"This interconnectivity of causes and impacts is the answer to the question 'Why should I care?' If we don't address the role we play in fuelling these disasters, then sooner or later they will arrive on everyone's doorstep," he told Al Jazeera.
Superficial solutions that address only the tip of the iceberg rather than the underlying root causes of disasters are bound to fail, the report emphasises. Half measures are also more likely to make the challenges worse and to bring additional risks.
"The answer has its roots in governance, inequality and personal attitudes and behaviour. It's connected to all of us," O'Connor said.
https://www.aljazeera.com/economy/2021/9/8/think-disasters-are-isolated-think-again-warns-un
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