Collaboration is a central theme to modern business, driving growth and innovation in key industries. For remote office scenarios, collaboration is critically important, and the following tips will allow you to get the most from remote employees and the resources you have available.
Software is the key to collaboration when working with remote employees. A poor software solution will lead to frustrating working conditions, a lack of productivity, and potential failure on daily tasks, regular deadlines, and large projects.
Collaborative software needs to be comprehensive and tailored to the needs of your business. What systems do your employees need to perform their work? For anything outside of basic office productivity, specialized applications and technology may be required. A managed IT solution is often the most cost-effective way to ensure that you are providing employees with the collaborative tools they need.
Managing performance becomes a unique challenge when working with remote employees.
Companies that embrace remote office structures can benefit from reduced overhead and higher employee job satisfaction, however, the benefits only matter if performance is not compromised. Setting targets, regular meetings, and both video and voice conferencing can help to maintain supervisory presence to keep employees motivated and compliant with expected professional standards.
In remote office environments data is exposed to more risk through negligence, theft, and fraud. For employees to collaborate and work to a high standard they need access to data and systems without unnecessary roadblocks. As an employer, it is your responsibility to closely control access without hindering productivity. Pre-screening and ongoing screening can help you to eliminate bad actors. Robust network and application level security solutions can protect from external data threats.
Training and utilizing technology correctly is perhaps the most important aspect of successful collaboration between remote employees and employers. Training ensures that employees maintain best security practices, that they know when and how to use specific applications, and that they are familiar with whatever storage and sharing platform is in place for collaborative workflow.
Remote office situations often require just as much training as a traditional workplace, and regular upskilling is necessary as technology changes. Training staff with an induction and continued upskilling program will ensure that everybody understands collaborative tools and how they are used.


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Niagara's agri-business sector
Thanks to a unique combination of deep sandy soils and favourable microclimates, Niagara is well known for its vineyards and tender fruit farms, but that's not the entire picture. Agri-business here extends far beyond the wine cellar and the fruit stand. The region also has a successful network of cash crop and dairy farms as well as greenhouses that specialize in fresh flowers, fruit, vegetables and licensed cannabis growth. The region has multiple agri-business processors and value-added agribusiness services that extend to support the entire industry.
Agriculture is a fundamental pillar of Niagara's economy thanks to its climate, land base, pool of experienced operators, research capability and skills that make it a leader in the production of agri-food products.
Niagara's agri-business sector is growing. In 2016, agriculture in Niagara contributed $1.41 billion to regional GDP and contributed 19,900 jobs – an increase of 15.5% over 2011. Niagara also represented 42.8% of the total agricultural economic impact of the Golden Horseshoe area, despite the fact that Niagara represents only 23% of its total farmland.
Niagara is home to over 1,800 farms and agricultural operations. With two-thirds of Ontario's tender fruit orchards located in the region, Niagara's orchards are responsible for the bulk of the province's peaches, cherries, pears, plums, prunes and grapes.
In addition to fruit farms, cash crops, greenhouses growers and poultry and egg producers are the most common type of operations found within the region.
Niagara is responsible for 90% of grape production in Ontario and is the largest wine producing region in Canada, making up 80% of Canada's total grape and wine production.
According to 2016 Canadian Census data, Niagara farms experienced a 16% increase in gross farm receipts between 2011 and 2016, with average farm revenues up by 47%. This is thanks in part to higher value crops as well as technology and automation creating higher yields per acre.
As of 2016, Niagara's agri-business sector was using just under 22 million square feet of greenhouse space in over 200 greenhouses. While greenhouses in Niagara have historically been focused on floriculture, there has been a 29% increase in vegetable production and a 57% increase in greenhouse other products, including licensed cannabis production for medicinal marijuana.
Niagara College's Agriculture and Environment Innovation Centre works with private and public sector partners to develop innovative solutions to address today's challenges in precision agriculture, GIS, computer programming, sustainable food production, plant growth, horticultural practices, greenhouse operations, aquaponics, environmental management and renewable energies.
Niagara College's Canadian Food and Wine Institute has 40 acres of on-site vineyards, a hop yard, culinary gardens, and a greenhouse to provide students with valuable hands-on farming experience and collaboration opportunities with local industry. The Innovation Centre team offers a full suite of services to support industry innovation and commercialization of new products and processes.
Brock University's Cool Climate Oenology and Viticulture Institute (CCOVI) has established itself as a successful, internationally recognized research institute that is producing important work focused on growing and harvesting cool climate grapes.
Brock University's Advanced Biomanufacturing Centre pairs the cutting-edge work of Brock's plant biologists and chemists with biotechnology companies that develop this research into innovative products and services. With its research strengths in plant biology and natural product chemistry, Brock's partnerships with Niagara's agricultural businesses are expected to create jobs and economic growth in Niagara and beyond.
Vineland Research and Innovation Centre is home to 20 on-site PhDs who are driving ground-breaking agricultural and horticultural research that is changing the way things are grown in Niagara and across the globe.
Niagara Parks Botanical Gardens and School of Horticulture in Niagara Falls was established in 1936 and is Canada's only residential school for Horticulture studies.
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redpajama_set_name: RedPajamaCommonCrawl


Check out our stock of Peugeot Boxer Engine for sale. Many Peugeot Boxer Engines in Stock inc 2.2 HDI, 3.0 HDI.
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Description: Delivery Of Equipment And Teaching Aids Under The Abc. I Am Learning Project.
Description: Delivery And Assembly Of Equipment And Furniture For Kindergarten In The School Complex In Ksiezy Ii.
Description: Educational Materials For Implementation School Program.
This section contains tender notices, project information, procurement news, procurement plans, contract awards from Poland pertaining to Education and Training sector. Click here to view the government, semi-government, PSU etc. Tender Notices according to the country sectors/categories.


text:
Q: Plymouth Boot Splash - Turn off switching to text I am working on an appliance system based on CentOS 6. I have created a custom boot splash, but I can't find any information on how I would disable the feature to switch to text mode (switching the splash on and off) through a keypress. I do not want a user to be able to drop the splash and view the boot messages. Can anyone point me in the right direction?

A: You should add plymouth quit to your rc.local file.  plymouth quit should do the trick..Here is a post of mine when I ran into a problem with plymouth: rc.local with read will not echo key strokes
I also found it useful to add nomodeset to the isolinux.cfg file
default linux
prompt 1
timeout 0
display boot.msg
F1 boot.msg
F2 options.msg
F3 general.msg
F4 param.msg
F5 rescue.msg
label linux
  kernel vmlinuz
  append initrd=initrd.img text ks=cdrom:/ks.cfg nousb nomodeset

Also, check out your ks.cfg file and make sure that rhgb quiet are NOT in the --apend= parm on your bootloader line.
CentOS 6.4 Kickstart bootloader problem

redpajama_set_name: RedPajamaStackExchange


text:
Mark Williams – Criminal Lawyer – Director
Potts Lawyers > Our Criminal & Litigation Lawyers > Mark Williams – Criminal Lawyer – Director
Mark is a director of the practice and holds a Bachelor of Laws and Bachelor of Business (Human Resource Management major) provide the grounding for a wide variety of criminal and quasi-criminal matters. He holds postgraduate qualifications in summary (Magistrates Court) prosecutions. Mark has also completed the Bar Practice Course and is qualified to practice as a Barrister; however his passion lies with hands-on dealings with clients and enjoys every moment of it. "I like the problem-solving aspect when helping clients…it's more rewarding and you don't get that as a Barrister".
Mark has experience in a variety of major criminal areas, and has developed a detailed knowledge of murder, weapons, child/adult sexual assault matters, internet/computer offences, fraud, and major drug trafficking / production & assault matters. Having investigated and prosecuted major crime – specifically in these areas; he now combines this with specific experience in criminal defence during the past decade to provide efficient and effective representation for clients who can look for someone they trust to secure the best possible result.
Mark is a compassionate listener and strategic, being able to think "outside the box" and consider all options. He thinks it's important to speak plain English and build solid relationships with clients so that they are involved in, and feel part of the process.
As a solicitor, Mark specialises in handling complex matters such as murder, weapons, drugs, fraud and serious assault. He recently successfully defended the "Surfers Paradise sniper". This was a high profile case and after 4 years of court battles, a jury acquitted our client. Even now, our client and his family still write to Mark each year, thanking him for his years of dedicated representation.
In 2016, Mark with the assistance of Danielle Warren represented another client who was charged with importing just over 1kg of cocaine into Australia. After years of tactical interlocutory steps a jury trial took place. The right Barrister was chosen and a full week of trial in the Supreme Court resulted in our client being acquitted by a jury on all charges. He has also represented families of workers in The royal Commission into the Home Insulation Program.
Mark explains that sometimes, despite contrary belief, the justice system does work and so long as you believe in your legal representative, have trust and faith, the right outcome will prevail.
Mark is a keen musician and studied classical piano musicianship for over 15 years. He also plays guitar, the Eb Tuba and other brass instruments. Mark has been in in several rock/pop cover bands that have played locally on the Gold Coast and is presently learning the cello, an instrument on his bucket list. He has taught himself the didgeridoo and the art of circular breathing.
As a member of the community, Mark is a member of the Gold Coast City Brass Band and is a committee member and Honorary Solicitor. Mark has devoted over 12 years of volunteer service to the State Emergency Service achieving the position of Assistant Local Controller Training – being responsible for the delivery of training to volunteers from within the community.
He appears regularly in all Queensland Courts and Crime Commission hearings. Mark regularly travels to regional areas in Queensland including Cairns, Townsville, Rockhampton, Gladstone, Maryborough, Central Queensland and the Sunshine Coast. Mark is in a position to travel to any area in order to represent his clients and with the heavy use of video/telephone facilities in regional areas; it is a very cost effective way to benefit by having an experienced and accessible lawyer.
Mark's background stems from a variety of major criminal areas, but for the past 10 years has developed a detailed knowledge of child/adult sexual assault matters, internet/computer offences, fraud, and major drug trafficking / production & assault matters. Having investigated major crime – specifically in these areas; he now combines this with specific experience in prosecutions to provide efficient and effective representation for clients who can look for someone they can trust to secure the best possible result.
He appears regularly in all Queensland Courts, Crime Commission hearings and Royal Commissions. Mark regularly travels to regional areas in Queensland bringing his experience to these areas. Presently, Mark has matters in Cairns, Townsville, Rockhampton, Gladstone, Maryborough, Central Queensland and the Sunshine Coast. Mark is in a position to travel to any area in order to represent his clients.
Click here for Mark's You Tube Channel Life of a Lawyer: https://www.youtube.com/channel/UC31Uu8jXkWRNMek26WxE1nQ
Click here for Mark's Linkedin profile.
Mark's Testimonials
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Nephrologists may oversee the treatment of dialysis patients.
Physicians who treat disorders and diseases of the kidneys are call nephrologists. Nephrology is a specialty of internal medicine, and nephrologists who meet the education and experience requirements may be certified by the American Board of Internal Medicine. These health care professionals work closely with urologists and other internal medicine physicians to treat kidney patients. Educational requirements for nephrologists are significantly higher than for general practice doctors, and their compensation is typically higher as well.
Nephrologists may be either medical doctors or doctors of osteopathy. Minimum educational requirements include a four-year bachelor's degree, four years of medical school and a three-year internal medicine residency, followed by at least two years in a nephrology fellowship. Average annual compensation for nephrologists ranges from $170,000 to $250,000, depending on such factors as experience, location and practice setup, according to the NephCareer website.
Nephrologists practice in the field of internal medicine. Wages for internal medicine specialists fell in the mid-range of other types of physicians as of 2010, according to the Bureau of Labor Statistics. Anesthesiologists earned the highest median annual wages at $407,292, while family practitioners who did not offer obstetric care received the lowest median annual compensation at $189,402. Internal medicine specialists, including nephrologists, earned median annual wages of $205,379.
The 2011 MedScape Nephrology Compensation Report placed the median compensation of nephrologists at $230,000. Approximately 10 percent of nephrologists surveyed reported earning less than $100,000 per year, while 30 percent reported a total compensation of more than $250,000. Salary is only one component of a nephrologist's total compensation package, which might also include bonuses, profit-sharing contributions and paid time off.
Approximately half of nephrologists reported no change in their annual compensation from 2009 to 2010, with 18 percent reporting a decline in income and 32 percent reporting an increase in income, according to the the 2011 MedScape Nephrology Compensation Report. Male nephrologists' tended to earn higher wages than their female counterparts, with annual compensation of $210,000 compared with $180,000. Compensation for nephrologists varied significantly based on the doctor's geographic location. Those who practiced in the Southwest earned a median compensation of $247,500, compared with $175,000 for their counterparts in the Northeast.
Healthy Children: What is a Pediatric Nephrologist?
Parker, Mike. "Kidney Doctor Salaries." Work - Chron.com, http://work.chron.com/kidney-doctor-salaries-1197.html. Accessed 25 April 2019.


text:
using System;
using System.Collections.Generic;
using System.ComponentModel;
using System.Data;
using System.Drawing;
using System.Linq;
using System.Text;
using System.Windows.Forms;
using Cheer.JsonVisualizer.CoreServices.Controls;

namespace Cheer.JsonVisualizer.UI
{
    internal partial class OptionsDialog
        : Dialog
    {
        public OptionsDialog()
        {
            InitializeComponent();
        }
    }
}

redpajama_set_name: RedPajamaGithub


Otto Lilienthal (23 Mey 1848 – 10 August 1896) wis a German pioneer o aviation who acame kent as the Glider Keeng.
↑ "Otto Lilienthal." Encyclopedia of World Biography, 2004. Retrieved: 7 January 2012.
This page wis last eeditit on 29 September 2015, at 22:40.


text:
Q: Python Scrapy function for i in Hello dear Stack overflow fellows!
I'm having some trouble in this for in function.
Here's the code:
  lista1 = pd.read_excel("Produtos_para_buscar.xlsx")
  for i in lista1:
        r = requests.get(f'https://www.petz.com.br/busca?q={i}', headers=headers, params=params)

        response = Selector(text=r.text)

        products = response.xpath('//li[@class="liProduct"]')

I want it to iterate the entire list and return each one of the values as a response.
But it's giving me only the first value of the list.
Any ideas?

A: When indexing a pandas dataframe it's best to treat it as a dictionary especially if you have multiple columns for the dataframe.
For example:
df = {'col1':[1, 2, 3], 'col2':[3, 2, 1]}
df_pd = pd.DataFrame(df)

"""This will get you the values for that column"""

for i in df_pd['col1']:
    print(i)

Furthermore, if you want a scrapy version for this:
class yourSpider(scrapy.Spider):
    name = 'your_spider'
    lista1 = pd.read_excel("Produtos_para_buscar.xlsx")
    start_urls = []
    for i in lista1['column_you_are_after']:
    start_urls.append(f'https://www.petz.com.br/busca?q={i}')

    def start_requests(self):
        for url in self.start_urls:
            yield scrapy.Request(
                url=url,
                callback = self.parse)

    def parse(self, response)
        products = response.xpath('.//li[@class="liProduct"]')
        for items in products:
            yield {
                'first_item':items.xpath('....')
  }

My guess is this is what you're after and it will likel work for your case. I noticed the xpath was a list so it's best to treat it like a list and that means to grab the values within a for-loop.


text:
\section{Introduction}

Let $(M,g)$ be a $4$-dimensional Riemannian manifold. For each plane $\sigma\subset T_pM$ at a point $p\in M$, denote by $\sigma^\perp$ the orthogonal plane to $\sigma$, i.e., $\sigma\oplus\sigma^\perp=T_pM$ is a $g$-orthogonal direct sum. Define the \emph{biorthogonal (sectional) curvature} of $\sigma$ as the average of the sectional curvatures of $\sigma$ and $\sigma^\perp$, i.e.,
\begin{equation*}
\sec^\perp_g(\sigma):=\tfrac12\big(\sec_g(\sigma)+\sec_g(\sigma^\perp)\big).
\end{equation*}
The Hopf Conjecture, that asks if $S^2\times S^2$ admits a metric with $\sec>0$, remains one of the most intriguing open problems in Riemannian geometry. With the standard product metric $g_0$, at every point $p\in S^2\times S^2$ there exists $\sigma\subset T_pM$ with $\sec^\perp_{g_0}(\sigma)=0$. Namely, any \emph{mixed plane} $\sigma$ at $p$ (i.e., spanned by vectors of the form $(X,0)$ and $(0,Y)$) is such that $\sigma^\perp$  is also a mixed plane, hence $\sec_{g_0}(\sigma)=\sec_{g_0}(\sigma^\perp)=0$.
A natural question in this context is if the weaker condition $\sec^\perp>0$ can be satisfied in $S^2\times S^2$ \cite{ezio}.

The goal of this note is to give a positive answer, also covering a stronger curvature positivity condition, that can be defined in any dimension. Namely, choose a distance (inducing the standard topology) on the Grassmannian bundle $Gr_2TM$ of planes tangent to $M$, and for each $\theta>0$ and $\sigma\subset T_pM$, let
\begin{equation*}
\sec^\theta_g(\sigma):=\min_{\substack{\sigma'\subset T_pM \\ \mathrm{dist}(\sigma,\sigma')\geq\theta}}\tfrac12\big(\sec_g(\sigma)+\sec_g(\sigma')\big).
\end{equation*}

\begin{thm}
For every $\theta>0$, there exist Riemannian metrics $g^\theta$ on $S^2\times S^2$ with $\sec^{\theta}_{g^\theta}>0$, arbitrarily close to the standard product metric $g_0$ in the $C^k$-topology, $k\geq0$. In particular, $S^2\times S^2$ admits metrics of positive biorthogonal curvature.
\end{thm}

The condition $\sec^\theta_g>0$ means that at every point $p\in M$, the average of sectional curvatures of any two planes $\sigma_1,\sigma_2\subset T_pM$ that are at least $\theta>0$ apart from each other is positive. One can intuitively think of $\theta$ as a lower bound for the ``angle'' between the planes considered. Notice that if $\theta_1<\theta_2$, then $\sec^{\theta_1}_g>0$ clearly implies $\sec^{\theta_2}_g>0$. Furthermore, for every metric $g$ on $M$, there exists $\theta_g>0$ such that if $\sec^{\theta}_g>0$ for some $0<\theta\leq\theta_g$, then $\operatorname{Ric}_g>0$, see Proposition~\ref{prop:ricci}. In particular, for $4$-manifolds, if $\theta\leq\min_{p\in M,\,\sigma\subset T_pM}\mathrm{dist}(\sigma,\sigma^\perp)$, then $\sec^\theta_g>0$ implies $\sec^\perp_g>0$.

The construction of $g^\theta$ is so that these metrics converge to a limit metric $g^0$ as $\theta\to 0$ (possibly different from the product metric), in the $C^k$-topology, for any $k\geq0$. This convergence easily implies that, for $\theta>0$ sufficiently small, the metrics $g^\theta$ have positive Ricci curvature (Proposition~\ref{prop:ricci}) and positive biorthogonal curvature. In particular, the above theorem shows that a natural interpolating condition between $\operatorname{Ric}>0$ and $\sec>0$ is satisfied on $S^2\times S^2$.

We stress that $\sec^\perp>0$ alone does not imply $\operatorname{Ric}>0$, as illustrated by $S^1\times S^3$ with the standard product metric. This metric clearly has $\sec^\perp>0$, but since $S^1\times S^3$ has infinite fundamental group, it does not support any metrics with $\operatorname{Ric}>0$. Thus, in order to have a condition of this type that is stronger than $\operatorname{Ric}>0$, it is crucial that $\sec^\theta>0$ can be satisfied no matter how small $\theta>0$, and that the corresponding metrics converge.
In general, $\sec^\perp>0$ only implies positive scalar curvature, which poses some topological restrictions on $4$-manifolds (e.g., vanishing of all the Seiberg-Witten invariants), but these restrictions are by far not as strong as the ones implied by $\sec>0$ or $\operatorname{Ric}>0$. In particular, although $\sec^\perp>0$ is comparatively flexible, generic smooth $4$-manifolds do not support metrics with this property. Another indication of this relative flexibility of $\sec^\perp>0$ is that $\mathds{C} P^2\#\overline{\mathds{C} P}^2$ also admits metrics with this property (Proposition~\ref{prop:cp2}); while, similarly to $S^2\times S^2$, it remains an open question whether it admits a metric with $\sec>0$.

Our metrics $g^\theta$ with $\sec^\theta>0$ on $S^2\times S^2$ can be chosen invariant under the antipodal action of $\mathds{Z}_2\oplus \mathds{Z}_2$. Thus, for all $\theta>0$, the quotient $\mathds{R} P^2\times \mathds{R} P^2$ also admits metrics with $\sec^\theta>0$, arbitrarily close to the standard product metric. This illustrates a remarkable difference between $\sec^\theta>0$ (in particular, $\sec^\perp>0$) and $\sec>0$ since, by Synge's Theorem, $\mathds{R} P^2\times \mathds{R} P^2$ cannot have a metric with $\sec>0$. It is, however, somewhat expected that obstructions of Synge type do not detect these average curvature conditions, since even finiteness of the fundamental group goes unnoticed.
We also remark that $(S^2\times S^2,g^\theta)$ has many points with planes of zero curvature (and even negative curvature), however any two such planes are always within distance $\theta$ from one another in the Grassmannian of planes tangent at that point.
In this way, $\theta$ corresponds to a measure of how big the regions formed by planes with nonpositive curvature can be in the Grassmannian. It would be interesting to know if metrics with $\sec^\theta>0$ on $S^2\times S^2$ can also be constructed while keeping $\sec\geq0$, as this could give a quantitative insight on the possibility of existence of quasipositively curved metrics.

The techniques used to construct all of the above metrics are (smooth) deformations.
Metric deformations to improve curvature have a long history, stemming from Berger and his students in the 1970's to the recent construction proposed by Petersen and Wilhelm \cite{pw,pwsphere} of a positively curved exotic sphere. Of particular importance in the present note are techniques developed by M\"uter~\cite{mueter} and Strake~\cite{strake,straket}, respectively regarding Cheeger deformations and deformations positive of first-order. The \emph{Cheeger deformation} is a method to attempt to increase curvature on nonnegatively curved manifolds with symmetries, by shrinking the metric in the direction of orbits of a large isometry group. This technique
was introduced by Cheeger~\cite{cheeger}, inspired by the construction of Berger metrics on odd-dimensional spheres, where the round metric is shrunk in the direction of the Hopf fibers. M\"uter~\cite{mueter} carried out a systematic study of Cheeger deformations in his PhD thesis under W. Meyer, establishing ground for a much better understanding of these deformed metrics. Strake~\cite{straket}, another PhD student of W. Meyer during the same period, studied metric deformations of nonnegatively curved metrics for which the first variation of the sectional curvature of any zero curvature plane is positive. These deformations are called \emph{positive of first-order}, and if the manifold is compact, they yield actual positively curved metrics. They also observed that, in this infinitesimal sense, Cheeger deformations are \emph{nonnegative} of first-order.

Our deformation process from the product metric $g_0$ to a metric with $\sec^\theta>0$ has two steps, in which the above techniques are combined. The first is a Cheeger deformation, described in detail by M\"uter~\cite{mueter,zillermueter}. More precisely, we consider the cohomogeneity one diagonal $\mathsf{SO}(3)$-action on $S^2\times S^2$ and shrink $g_0$ in the direction of the orbits. This deformation gives a family of metrics $g_t$, $t>0$, with $\sec_{g_t}\geq0$ and much fewer planes of zero curvature than $g_0$. Namely, $(S^2\times S^2,g_t)$ has a circle's worth of zero curvature planes on points that lie on the diagonal or the anti-diagonal $\pm\Delta S^2=\{(p,\pm p):p\in S^2\}\subset S^2\times S^2$, and a unique zero curvature plane at any other point. This means that $\sec^\theta\geq0$, and equality holds only for some planes whose base point is in one of the submanifolds $\pm\Delta S^2$ (Proposition~\ref{prop:geomcheeger}).
Next, for fixed $t>0$, set $g:=g_t$. The second step is to employ a first-order local conformal deformation $g_s=g+s\,h$, where $h=\phi\, g$, and $\phi$ is supported in a tubular neighborhood of $\pm\Delta S^2$. Given the geometry of $(S^2\times S^2,g)$, we construct $\phi$ such that the first derivative with respect to $s$ of the average of two $g_s$-sectional curvatures is positive (Proposition~\ref{prop:final}). The function $\phi$ is proportional to the squared $g$-distance to $\pm\Delta S^2$, multiplied by a cutoff function. The strategy for such a construction is adapted from Strake~\cite{strake,straket}. Finally, a standard compactness argument (Proposition~\ref{prop:cpct}) implies that $\sec^\theta_{g_s}>0$ for all sufficiently small $s>0$, proving the desired result.

This paper is organized as follows. In Section~\ref{sec:cheeger}, we review basic aspects of Cheeger deformations, following M\"uter~\cite{mueter,zillermueter}. We describe the metric on $S^2\times S^2$ obtained by a Cheeger deformation with respect to the diagonal $\mathsf{SO}(3)$-action in terms of $\sec^\theta$. In Section~\ref{sec:firstorder}, we analyze the effects of a first-order deformation and construct the variation starting from the Cheeger deformed metric that proves the above Theorem. Some remarks on the geometry of the constructed metrics are given in Section~\ref{sec:remarks}. Finally, we briefly discuss $4$-manifolds with positive biorthogonal curvature (including the construction for $\mathds{C} P^2\#\overline{\mathds{C} P}^2$) in Section~\ref{sec:4mflds}.

\smallskip
\noindent
{\bf Acknowledgement.} It is a pleasure to thank Fernando Galaz-Garc\'ia, Karsten Grove, Paolo Piccione and Wolfgang Ziller for valuable comments and suggestions during the elaboration of this paper. We also express our sincere gratitude to the referee for the careful reading of the manuscript and constructive criticism.

\section{First step: Cheeger deformation}\label{sec:cheeger}

Although the techniques used in this section are mostly available elsewhere in the literature, see \cite{mueter,zillermueter,zillersurvey}, we briefly recall a few basic aspects as a service to the reader. For convenience, we use the same notation as the above references.

\subsection{Cheeger deformation}
Let $(M,g)$ be a Riemannian manifold and $\mathsf G$ a compact Lie group that acts on $M$ by isometries. The \emph{Cheeger deformation} of $g$ is a $1$-parameter family $g_t$, $t\geq0$, of $\mathsf G$-invariant metrics on $M$, defined as follows. Let $Q$ be a bi-invariant metric on $\mathsf G$, and endow $M\times\mathsf G$ with the product metric $g+\tfrac1t Q$. Consider the submersion
\begin{equation}
\rho\colon M\times\mathsf G\to M, \quad \rho(p,h)=h^{-1}p,
\end{equation}
and define $g_t$ as the metric on $M$ that turns $\rho$ into a Riemannian submersion. The family of metrics $g_t$ extends smoothly across $t=0$, with $g_0=g$, thus providing a deformation of such metric. Since $\sec_Q\geq0$, it follows immediately from the Gray-O'Neill formula that, if $\sec_{g_0}\geq0$, then also $\sec_{g_t}\geq0$, $t\geq0$. As we will see, many planes with zero curvature with $g_0$ usually gain positive curvature with $g_t$.

For each $p\in M$, denote by $\mathsf G_p$ the isotropy group at $p$ and by $\mathfrak g_p$ its Lie algebra. Fix the $Q$-orthogonal splitting $\mathfrak g=\mathfrak g_p\oplus\mathfrak m_p$, and identify $\mathfrak m_p$ with the tangent space $T_p\mathsf G(p)$ to the $\mathsf G$-orbit through $p$ via action fields. More precisely, we identify $X\in\mathfrak m_p$ with $X^*_p=\frac{\mathrm{d}}{\mathrm{d} s}\exp(sX)p\big|_{s=0}\in T_p\mathsf G(p)$. This determines a $g_t$-orthogonal splitting
$T_pM=\v_p\oplus\mathcal{H}_p$ in \emph{vertical space} $\v_p:=T_p\mathsf G(p)=\{X^*_p:X\in\mathfrak m_p\}$ and \emph{horizontal space} $\mathcal{H}_p:=\{v\in T_pM : g_t(v,\v_p)=0\}$. Notice that the dimensions of $\v_p$ and $\mathcal{H}_p$ may vary with $p\in M$, hence these are not distributions.

Let $P_t\colon\mathfrak m_p\to\mathfrak m_p$ be the $Q$-symmetric automorphism that relates the metrics $Q$ and $g_t$, i.e., such that
\begin{equation}\label{eq:p}
Q(P_t(X),Y)=g_t(X^*_p,Y^*_p), \quad X,Y\in\mathfrak m_p.
\end{equation}
It is an easy computation that $P_t$ is determined by $P_0$ in the following way:
\begin{equation}\label{eq:pt}
P_t=(P_0^{-1}+t\operatorname{Id})^{-1}=P_0\,(\operatorname{Id}+tP_0)^{-1}, \quad t\geq0,
\end{equation}
see \cite[Prop 1.1]{zillermueter}.
Thus, if we let $C_t\colon T_pM\to T_pM$ be the $g$-symmetric automorphism that relates $g$ and $g_t$, i.e., such that
\begin{equation}\label{eq:ctdef}
g(C_t(X),Y)=g_t(X,Y), \quad X,Y\in T_pM,
\end{equation}
we then get
\begin{equation}\label{eq:ct}
C_t(X)=P_0^{-1}P_t(X^\v)+X^\mathcal{H}, \quad X\in T_pM,
\end{equation}
where $X^\v$ and $X^\mathcal{H}$ are the vertical and horizontal components of $X$ respectively. This reveals how the geometry of $g_t$ changes with $t$, since if $P_0$ has eigenvalues $\lambda_i$, then $C_t$ has eigenvalues $\frac{1}{1+t\lambda_i}$ corresponding to the vertical directions and eigenvalues $1$ in the horizontal directions. In other words, as $t$ grows, the metric $g_t$ shrinks in the direction of the orbits and remains the same in the orthogonal directions.

\subsection{Curvature evolution}
Let us now analyze how the curvature changes under this deformation. Henceforth, we assume the initial metric $g_0$ has $\sec_{g_0}\geq0$. As explained above, this implies $\sec_{g_t}\geq0$ for all $t\geq0$. Given $X\in T_pM$, denote by $X_{\mathfrak m}$ the unique vector in $\mathfrak m_p$ such that $(X_{\mathfrak m})^*_p=X_p^\v$. Also, given a plane $\sigma=\operatorname{span}\{X,Y\}$, we write
\begin{equation*}
C_t^{-1}(\sigma):=\operatorname{span}\{C_t^{-1}X,C_t^{-1}Y\}.
\end{equation*}
As explained by Ziller \cite{zillermueter}, the crucial observation of M\"uter is that, to analyze the evolution of $\sec_{g_t}$, it is much more convenient to study $\sec_{g_t}(C_t^{-1}(\sigma))$ rather than $\sec_{g_t}(\sigma)$. In more recent literature, the $1$-parameter family of bundle automorphisms induced by $C_t^{-1}$ in the Grassmannian bundle $Gr_2TM$ of planes on $M$ is being called \emph{Cheeger reparametrization}, see \cite{pw,pwsphere}. The following result of M\"uter~\cite[Satz 3.10]{mueter} (see also \cite[Cor 1.4]{zillermueter}) summarizes how the curvature of $g_t$ evolves.

\begin{proposition}\label{prop:mueter}
Let $g_t$ be the Cheeger deformation of $g_0$, $\sec_{g_0}\geq0$. Given $\sigma=\operatorname{span}\{X,Y\}\subset T_pM$, consider the unnormalized $g_t$-sectional curvature of $C_t^{-1}(\sigma)$:
\begin{equation*}
k_c(t):=\big\|C_t^{-1}X\wedge C_t^{-1}Y\big\|^2_{g_t}\sec_{g_t}(C_t^{-1}(\sigma))=g_t\big(R_t(C_t^{-1}X,C_t^{-1}Y)C_t^{-1}Y,C_t^{-1}X\big),
\end{equation*}
where $R_t$ is the curvature tensor of $g_t$. If $\sec_{g_0}(\sigma)=0$, then:
\begin{itemize}
\item[(i)] $k'_c(0)\geq0$;
\item[(ii)] If $k'_c(0)=0$ and $[P_0 X_{\mathfrak m},P_0Y_{\mathfrak m}]\neq0$, then $k_c''(0)=0$, $k_c'''(0)>0$ and $k_c(t)>0$ for all $t>0$;
\item[(iii)] If $k'_c(0)=0$ and $[P_0 X_{\mathfrak m},P_0 Y_{\mathfrak m}]=0$, then $k_c(t)=0$ for all $t>0$.
\end{itemize}
In particular, if $[P_0 X_{\mathfrak m}, P_0 Y_{\mathfrak m}]\neq0$, i.e., the plane $\operatorname{span}\{P_0X_{\mathfrak m},P_0Y_{\mathfrak m}\}$ has positive curvature in $(\mathsf G,Q)$, and $\sec_{g_0}(\sigma)=0$, then $\sec_{g_t}(C_t^{-1}(\sigma))>0$ for all $t>0$.
\end{proposition}

Observe that, from (iii), if $\sigma$ is tangent to a totally geodesic flat torus in $M$ that contains a horizontal direction, then $\sec_{g_t}(C_t^{-1}(\sigma))=0$, $t\geq0$, i.e., $\sigma$ remains flat. On the other hand, we also get the following important positive result:

\begin{corollary}\label{cor:so3}
Assume $\mathsf G=\mathsf{SO}(3)$ or $\mathsf G=\mathsf{SU}(2)$, so that $\sec_Q>0$. If $\sec_{g_0}(\sigma)=0$ and the image of the projection of $\sigma\subset\v_p\oplus\mathcal{H}_p$ onto $\v_p$ is $2$-dimensional, then $\sec_{g_t}(C_t^{-1}(\sigma))>0$ for all $t>0$. In other words, up to the Cheeger reparametrization, zero curvature planes with nondegenerate vertical projection have positive curvature with $g_t$, for all $t>0$.
\end{corollary}

\subsection{The case of $S^2\times S^2$}
Consider $S^2\times S^2$ endowed with the standard product metric $g_0$ and the diagonal $\mathsf{SO}(3)$-action:
\begin{equation*}
A\cdot(p_1,p_2)=(A\,p_1,A\,p_2), \quad p=(p_1,p_2)\in S^2\times S^2\subset\mathds{R}^3\oplus\mathds{R}^3, A\in\mathsf{SO}(3).
\end{equation*}
This is a cohomogeneity one isometric action with orbit space a closed interval, so there are codimension one principal orbits (corresponding to interior points of the interval) and two singular orbits (corresponding to the endpoints), see \cite{zillersurvey}. These singular orbits are the \emph{diagonal} and \emph{anti-diagonal} submanifolds:
\begin{equation*}
\pm\Delta S^2:=\{(p_1,\pm p_1):p_1\in S^2\}\subset S^2\times S^2.
\end{equation*}
The principal isotropy $\mathsf G_p$, $p\not\in\pm\Delta S^2$, is trivial, since it consists of orientation-preserving isometries of $\mathds{R}^3$ that fix two linearly independent directions. The singular isotropies are formed by orientation-preserving isometries of $\mathds{R}^3$ that fix one direction, hence are isomorphic to $\mathsf{SO}(2)$. Thus, the group diagram of this action is $\{1\}\subset\{\mathsf{SO}(2),\mathsf{SO}(2)\}\subset\mathsf{SO}(3)$.

Following M\"uter~\cite{mueter}, we identify the Lie algebra of $\mathsf{SO}(3)$ with $\mathds{R}^3$ by:
\begin{equation*}
\mathfrak{so}(3)\ni Z=\left(\begin{array}{ccc}
0 & -z_3 & z_2 \\
z_3 & 0 & -z_1 \\
-z_2 & z_1 & 0
\end{array}\right)\longleftrightarrow\left(\begin{array}{c}
z_1 \\ z_2 \\ z_3\end{array}\right)=z\in\mathds{R}^3.
\end{equation*}
Considering $(\mathfrak{so}(3),Q)$ endowed with the standard bi-invariant metric, the above is an isometric identification with Euclidean space $(\mathds{R}^3,\langle\cdot,\cdot\rangle)$. In this way, since the Lie exponential in $\mathsf{SO}(3)$ is given by matrix exponentiation, the action field induced by $Z\in\mathfrak{so}(3)$ is:
\begin{equation}\label{eq:actionfield}
Z^*_p=(Z^*_{p_1},Z^*_{p_2}) =(Z\,p_1,Z\,p_2)=(z\wedge p_1,z\wedge p_2)\in T_p(S^2\times S^2).
\end{equation}
So, if $x,y\in\mathds{R}^3$ are such that $\langle x,p_1\rangle=\langle y,p_2\rangle=0$, then for all $z\in\mathds{R}^3$,
\begin{equation}\label{eq:coiso}
g_0\big((X^*_{p_1},Y^*_{p_2}),Z^*_p\big)=\langle x\wedge p_1,z\wedge p_1\rangle +\langle y\wedge p_2,z\wedge p_2\rangle =\langle x+y,z\rangle.
\end{equation}
Thus, the vector $(X^*_{p_1},-X^*_{p_2})\in T_p(S^2\times S^2)$ is horizontal, whenever $x\in\{p_1,p_2\}^\perp:=\{x\in\mathds{R}^3:\langle x,p_1\rangle=\langle x,p_2\rangle=0\}$. By dimensional reasons, it then follows that the horizontal space for the $\mathsf{SO}(3)$-action on $S^2\times S^2$ at $p=(p_1,p_2)$ is
\begin{equation*}
\mathcal{H}_p=\big\{(X^*_{p_1},-X^*_{p_2})\in T_p(S^2\times S^2):x\in\{p_1,p_2\}^\perp\big\}.
\end{equation*}
Recall that the vertical space is $\v_p=\{(X^*_{p_1}, X^*_{p_2}):X\in\mathfrak m_p\}$, see \eqref{eq:actionfield}. For general $x,y\in\mathds{R}^3$, analogously to \eqref{eq:coiso}, we have:
\begin{multline*}
g_0\big(X^*_{p},Y^*_{p}\big)=g_0\big((X^*_{p_1},X^*_{p_2}),(Y^*_{p_1},Y^*_{p_2})\big)=\langle x\wedge p_1,y\wedge p_1\rangle +\langle x\wedge p_2,y\wedge p_2\rangle =\\
\langle p_1\wedge(x\wedge p_1),y\rangle+\langle p_2\wedge(x\wedge p_2),y\rangle =\big\langle (2x-\langle x,p_1\rangle p_1-\langle x,p_2\rangle p_2),y\big\rangle.
\end{multline*}
From \eqref{eq:p}, the above is equal to $\langle P_0\, X,Y\rangle$, so we get an explicit formula for $P_0\colon\mathfrak m_p\to\mathfrak m_p$ in our example:
\begin{equation*}
P_0\, X=2X-\langle X,p_1\rangle p_1-\langle X,p_2\rangle p_2.
\end{equation*}
In particular, it follows that the subspace $\{p_1,p_2\}^\perp\subset \mathfrak m_p$ is invariant under $P_0$ and hence under $P_t$ and $C_t$, see \eqref{eq:pt} and \eqref{eq:ct}.

Let $\pi\colon\{p_1,p_2\}^\perp\to\{p_1,p_2\}^\perp/\sim$ be the projection onto the corresponding real projective space. For each $x\in\{p_1,p_2\}^\perp$, consider the vertizontal\footnote{i.e., this plane is spanned by one vertical and one horizontal vector.} plane
\begin{equation}\label{eq:sigmapi}
\sigma_{\pi(x)}:=\operatorname{span}\{(X^*_{p_1},0),(0,X^*_{p_2})\}=\operatorname{span}\{(X^*_{p_1},X^*_{p_2}),(X^*_{p_1},-X^*_{p_2})\}.
\end{equation}
This is the unique mixed plane at $p$ that contains the horizontal vector $(X^*_{p_1},-X^*_{p_2})$. Thus, from Corollary~\ref{cor:so3}, $\{\sigma_{\pi(x)}\subset T_p(S^2\times S^2):x\in\{p_1,p_2\}^\perp\}$ are the only $g_0$-flat planes at $p$ that remain $g_t$-flat for $t>0$, up to the Cheeger reparametrization. As a matter of fact, by the above, the Cheeger reparametrization fixes such planes, i.e.,
\begin{equation}\label{eq:ctfix}
C_t^{-1}(\sigma_{\pi(x)})=\sigma_{\pi(x)}.
\end{equation}

In conclusion, for any $t>0$, $\sec_{g_t}\geq0$ and $\sec_{g_t}(\sigma)=0$ if and only if $\sigma=\sigma_{\pi(x)}\subset T_p(S^2\times S^2)$ for some $x\in\{p_1,p_2\}^\perp$. In particular, $g_t$-flat planes at $p$ are parametrized by $\pi(\{p_1,p_2\}^\perp)$. Thus, in $(S^2\times S^2,g_t)$, $t>0$, there is a circle's worth of zero curvature planes at each $p\in\pm\Delta S^2$, a unique zero curvature plane at each $p\not\in\pm\Delta S^2$, and all other planes have positive curvature. In terms of $\sec^\theta$, we thus have the following.

\begin{proposition}\label{prop:geomcheeger}
Let $g_t$ be the Cheeger deformation of $(S^2\times S^2,g_0)$ with respect to the diagonal $\mathsf{SO}(3)$-action.
Then, for any $\theta>0$ and $t>0$, $\sec^\theta_{g_t}\geq0$ and equality only holds for planes at some $p=(p_1,\pm p_1)\in\pm\Delta S^2$. If $\sigma\subset T_{p}(S^2\times S^2)$ has $\sec^\theta_{g_t}(\sigma)=0$, then $\sigma=\sigma_{\pi(x)}$ for some $x\in\{p_1\}^\perp$; in particular, $\sigma$ is not tangent to the submanifold $\pm\Delta S^2$.
\end{proposition}

\begin{proof}
The above statements follow immediately from M\"uter~\cite[Satz 4.26]{mueter} (see also Ziller~\cite[p. 5]{zillermueter} and Kerin~\cite[Rem 4.3]{kerin}), as well as from the above discussion.
\end{proof}

\begin{remark}
For $n\geq3$, although there exists an analogous cohomogeneity one $\mathsf{SO}(n+1)$-action on $S^n\times S^n$, the corresponding Cheeger deformation fails to produce so many positively curved planes. This is due to the fact that $\mathsf{SO}(n+1)$, $n\geq 3$, is not positively curved, cf. Corollary~\ref{cor:so3}. As a result, this step in the construction of our metrics with $\sec^\theta>0$ only works on $S^n\times S^n$ if $n=2$.
\end{remark}

\section{Second step: first-order local conformal deformation}\label{sec:firstorder}

As seen above, the Cheeger deformed metrics $g_t$, $t>0$, have $\sec^\theta_{g_t}\geq0$ and equality holds only for certain planes (of the form \eqref{eq:sigmapi}) at $\pm\Delta S^2$. In order to get these planes to also have $\sec^\theta>0$, we now carry out a (local) first-order conformal deformation, inspired by results of Strake~\cite{strake}. More precisely, choose $g$ to be a Cheeger deformed metric $g_t$ for any $t>0$ and consider the new $1$-parameter family
\begin{equation}\label{eq:gs}
g_s:=g+s\,h, \quad s\in\,]-\varepsilon,\varepsilon[,
\end{equation}
where $h$ is some symmetric $(0,2)$-tensor to be defined, and $\varepsilon>0$ is small enough so that $g_s$ is still a Riemannian metric. Given the above geometry of the Cheeger deformed metric $g$, we will choose $h$ such that
\begin{multline}\label{eq:goal}
\frac{\mathrm{d}}{\mathrm{d} s}\big(\sec_{g_s}(\sigma_1)+\sec_{g_s}(\sigma_2)\big)\Big|_{s=0}>0, \quad \mbox{ for all } \sigma_1,\sigma_2\subset T_p(S^2\times S^2) \mbox{ with }\\ p\in\pm\Delta S^2 \mbox{ and }\sec_g(\sigma_1)=\sec_g(\sigma_2)=0.
\end{multline}
The crucial observation that makes this possible is that these planes are never tangent to $\pm\Delta S^2$. Our choice will be such that $h$ is supported only near $\pm\Delta S^2$ and is pointwise proportional to $g$, justifying the terminology.
We start by recalling the first variation of $\sec_{g_s}(\sigma)$, see Strake~\cite[Sec 3.a]{strake}.

\begin{proposition}\label{prop:secondstep}
Let $(M,g)$ be a Riemannian manifold with $\sec_g\geq0$ and $X,Y\in T_pM$ be $g$-orthonormal vectors that span a $g$-flat plane $\sigma\subset T_pM$. Consider a first-order variation $g_s=g+s\,h$. Then the first variation of $\sec_{g_s}(\sigma)$ is given by
\begin{equation*}
\frac{\mathrm{d}}{\mathrm{d} s}\sec_{g_s}(\sigma)\Big|_{s=0}=\nabla_X\nabla_Y h(X,Y)-\tfrac12\nabla_X\nabla_X h(Y,Y)-\tfrac12\nabla_Y\nabla_Y h(X,X).
\end{equation*}
In particular, if $h=\phi\,g$, then
\begin{equation}\label{eq:dds}
\frac{\mathrm{d}}{\mathrm{d} s}\sec_{g_s}(\sigma)\Big|_{s=0} =-\tfrac12 \operatorname{Hess}\phi\,(X,X)-\tfrac12\operatorname{Hess}\phi\,(Y,Y).
\end{equation}
\end{proposition}

Now, observe that if $N\subset M$ is an embedded submanifold, the squared distance function $\psi(p)=\mathrm{dist}_g(p,N)^2$ is smooth in a sufficiently small tubular neighborhood of $N$. The gradient of $\psi$ at $p$ vanishes if $p\in N$, and points in the outward radial direction if $p\not\in N$. The Hessian of $\psi$ at $p\in N$ is given by:
\begin{equation}\label{eq:hess}
\operatorname{Hess}\psi\,(X,X)=2g(X_\perp,X_\perp)=2\|X_\perp\|^2_g, \quad X\in T_pM,
\end{equation}
where $X=X_\top+X_\perp\in T_pN\oplus (T_pN)^\perp$ is the $g$-orthogonal decomposition in tangent and normal parts to $N$.

\begin{proposition}\label{prop:final}
Consider the metrics $g_s$ on $S^2\times S^2$, given by \eqref{eq:gs}. There exists a smooth function $\phi\colon S^2\times S^2\to\mathds{R}$, supported in a neighborhood of $\pm\Delta S^2$, such that if $h=\phi\,g$, then \eqref{eq:goal} holds.
\end{proposition}

\begin{proof}
From Proposition~\ref{prop:geomcheeger}, the only planes $\sigma$ with $\sec^\theta_g(\sigma)=0$ are of the form $\sigma_{\pi(x)}\subset T_{(p_1,\pm p_1)}(S^2\times S^2)$ for some $x\in\{p_1\}^\perp$. These planes are vertizontal, i.e., they contain a direction normal to $\pm\Delta S^2$.

Let us analyze the plane $\sigma_{\pi(x)}$  at $(p_1,p_1)\in\Delta S^2$, the case of $(p_1,-p_1)\in-\Delta S^2$ being totally analogous. As mentioned above, the function $\psi_+(p)=\mathrm{dist}_g(p,\Delta S^2)^2$ is smooth in a tubular neighborhood $D(\Delta S^2)$. Let $\chi_+\colon S^2\times S^2\to\mathds{R}$ be a smooth cutoff function that vanishes outside $D(\Delta S^2)$ and is equal to $1$ in a smaller tubular neighborhood of $\Delta S^2$.
According to \eqref{eq:sigmapi}, \eqref{eq:dds} and \eqref{eq:hess}, if we set $h_+=-(\chi_+\psi_+)\,g$ and $g^+_s=g+s\,h_+$, then
\begin{equation*}
\frac{\mathrm{d}}{\mathrm{d} s}\sec_{g^+_s}(\sigma_{\pi(x)})\Big|_{s=0}=\big\|(X^*_{p_1},-X^*_{p_1})\big\|^2_g>0.
\end{equation*}
Defining $\psi_-$ and $\chi_-$ analogously, we get $h_-=-(\chi_-\psi_-)\,g$ and $g^-_s=g+s\,h_-$ with the same property as above for planes at $(p_1,-p_1)\in -\Delta S^2$ with zero $g$-curvature. Thus, the function $\phi:=-(\chi_+\psi_+)-(\chi_-\psi_-)$ has the desired properties. More precisely, $h=\phi\,g=h_+ +h_-$ is such that $g_s=g+s\,h$ coincides with $g_s^\pm$ near $\pm\Delta S^2$. Hence $\frac{\mathrm{d}}{\mathrm{d} s}\sec_{g_s}(\sigma)\big|_{s=0}>0$ for all planes $\sigma$ at $\pm\Delta S^2$ such that $\sec_g(\sigma)=0$; in particular, \eqref{eq:goal} holds.
\end{proof}

In order to conclude the proof of the Theorem in the Introduction, we quote the following elementary fact.

\begin{proposition}\label{prop:cpct}
Let $f\colon \,]-\varepsilon,\varepsilon[\,\times K\to\mathds{R}$ be a smooth function, where $K$ is a compact manifold (possibly with boundary). Assume that $f(0,x)\geq 0$ for all $x\in K$ and $\frac{\partial f}{\partial s}(0,x)>0$ if $f(0,x)=0$. Then, there exists $s_*>0$ such that $f(s,x)>0$ for all $x\in K$ and $0<s<s_*$.
\end{proposition}

\begin{proof}
Follows by a routine compactness argument, using the Taylor polynomial of $f(s,x)$ at $s=0$.
\end{proof}

Consider the following compact subset of the fibered product of two copies of the Grassmannian bundle $Gr_2TM$, where $M=S^2\times S^2$:
\begin{equation*}
K_\theta:=\big\{(p,\sigma_1,\sigma_2)\in M\timesGr_2TM\timesGr_2TM : \sigma_1,\sigma_2\subset T_pM, \,\mathrm{dist}(\sigma_1,\sigma_2)\geq\theta\big\}.
\end{equation*}
Due to \eqref{eq:goal} and $\theta>0$, the function
\begin{equation*}
f\colon \,]-\varepsilon,\varepsilon[\,\times K_\theta\longrightarrow\mathds{R}, \quad f(s,\sigma_1,\sigma_2)=\tfrac12\big(\sec_{g_s}(\sigma_1)+\sec_{g_s}(\sigma_2)\big),
\end{equation*}
satisfies the hypotheses of Proposition~\ref{prop:cpct}, hence there exists $s_\theta>0$ such that $f(s,\sigma_1,\sigma_2)>0$ for all $(\sigma_1,\sigma_2)\in K_\theta$ and $0<s<s_\theta$. This is equivalent to $\sec^\theta_{g_s}>0$, for $0<s<s_\theta$; so we can finally choose $g^\theta:=g_s$, for any $0<s<s_\theta$. Moreover, it follows immediately from the above results (Propositions~\ref{prop:geomcheeger}, \ref{prop:secondstep} and \ref{prop:cpct}) that  $g^\theta$ can be constructed arbitrarily close to the standard product metric $g_0$, in any $C^k$-topology, $k\geq0$, by choosing $t>0$ (the duration of the Cheeger deformation) and $s>0$ sufficiently small. This concludes the proof of the Theorem in the Introduction.

\section{Remarks on the construction}\label{sec:remarks}

\subsection{First-order deformations and the Hopf conjecture}
The above first-order deformation $g_s$ works to get $\sec^\theta>0$ on all of $M=S^2\times S^2$ because the only points $p\in M$ that have planes $\sigma_1,\sigma_2\subset T_pM$ with $f(0,\sigma_1,\sigma_2)=0$ are contained in the submanifolds $\pm\Delta S^2$, which admit a relatively compact neighborhood and where $\frac{\partial f}{\partial s}(0,\sigma_1,\sigma_2)>0$. The same cannot be done for the sectional curvature because at \emph{every point} there is a plane $\sigma$ with $\sec_g(\sigma)=0$. The only type of first-order deformation that would give $\sec_{g_s}>0$ would be one with $\frac{\mathrm{d}}{\mathrm{d} s}\sec_{g_s}(\sigma)\big|_{s=0}>0$ for all $\sigma$ with $\sec(\sigma)=0$. It was proved by Strake~\cite[Prop. 4.3]{strake} that such a deformation does not exist on $(S^2\times S^2,g)$, due to the presence of totally geodesic flat tori.

\subsection{Other compact subsets}\label{subsec:othercpcts}
Notice that condition \eqref{eq:goal} does not contain any information on the compact subset $K_\theta$, which is the domain considered for the function $f(s,\sigma_1,\sigma_2)=\tfrac12\big(\sec_{g_s}(\sigma_1)+\sec_{g_s}(\sigma_2)\big)$. This means that the same argument above could be applied to obtain positivity of the average of sectional curvatures of planes that satisfy some other conditions codified in the form of a compact subset
\begin{equation*}
K\subset \big\{(p,\sigma_1,\sigma_2)\in M\timesGr_2TM\timesGr_2TM : \sigma_1,\sigma_2\subset T_pM\big\},
\end{equation*}
where $M=S^2\times S^2$. Replacing the domain of $f$ by $K$, provided that $K$ does not intersect the diagonal (i.e., the subset $\Delta=\{(p,\sigma,\sigma):\sigma\subset T_pM\}$), we get from Proposition~\ref{prop:cpct} that $f(s,\sigma_1,\sigma_2)>0$ for $s>0$ small enough and all $(\sigma_1,\sigma_2)\in K$. We must require that $K$ be away from the diagonal, otherwise $f(0,\sigma_1,\sigma_2)$ would also have zeros on points outside the singular orbits $\pm\Delta S^2$, and there is no first-order variation that accounts for $\frac{\partial f}{\partial s}(0,\sigma_1,\sigma_2)>0$ at all such points. Notice also that for every $K$ with the required properties above, there exists $\theta>0$ such that $K\subset K_\theta$, so all other possibilities are accounted for by using the domains $K_\theta$.

\subsection{Ricci curvature}\label{subsec:ricci}
Since we know that $g^\theta$ can be constructed arbitrarily $C^k$-close (for any $k\geq0$) to the product metric $g_0$, it automatically follows that such metrics can be chosen with positive Ricci curvature. Nevertheless, existence of metrics with $\operatorname{Ric}>0$ can be directly deduced from the existence of metrics with $\sec^\theta>0$ for arbitrarily small $\theta>0$, that converge to a limit metric as $\theta\to0$, in the $C^k$-topology, $k\geq0$, as we shall now prove. This abstract property is hence stronger\footnote{Notice that the same is not true for $\sec^\perp>0$ on $4$-manifolds, as illustrated by $S^1\times S^3$ with the standard product metric. The crucial point is that $\sec^\theta>0$ has to be satisfied no matter how small $\theta>0$, and the metrics that do so do not diverge or degenerate.} than $\operatorname{Ric}>0$ for compact manifolds (and, of course, weaker than $\sec>0$), regardless of the dimension of $M$. In this way, the Theorem in the Introduction shows that a natural interpolating condition between $\operatorname{Ric}>0$ and $\sec>0$ is satisfied on $S^2\times S^2$.

\begin{proposition}\label{prop:ricci}
Let $M$ be a compact $n$-dimensional manifold such that for every $\theta>0$ there exists a metric $g^\theta$ with $\sec^\theta_{g^\theta}>0$. Assume that there exists a metric $g^0$ on $M$ such that $g^\theta\to g^0$ in the $C^0$-topology, as $\theta\to 0$. Then $\operatorname{Ric}_{g^\theta}>0$ for $\theta>0$ sufficiently small; in particular, if $g^\theta\to g^0$ also in the $C^2$-topology, then $\operatorname{Ric}_{g^0}\geq0$.\footnote{Moreover, in this case, one can also easily prove that $\sec_{g^0}\geq0$.}
\end{proposition}

\begin{proof}
For any metric $g$ on $M$, define:
\begin{equation}
\theta_g:=\min_{p\in M}\left(\min_{\substack{v\in T_pM, \\ g(v,v)=1}}\left(\min_{\substack{w_1,w_2\in T_pM, \\ g(v,w_j)=0, \\ g(w_i,w_j)=\delta_{ij}}} \mathrm{dist}\Big(\operatorname{span}\{v,w_1\},\operatorname{span}\{v,w_2\}\Big)\right)\right).
\end{equation}
The above defines a positive number, that depends continuously on the metric $g$, such that if $\sec^{\theta}_g>0$ for some $0<\theta\leq\theta_g$, then $\operatorname{Ric}_g>0$. In fact, $\operatorname{Ric}_g(v)>0$ for any direction $v$, since this is a sum of $(n-1)$ sectional curvatures whose pairwise average is positive, because $\sec^\theta_g>0$.

Given the continuous family $\mathcal G:=\{g^\theta:\theta\in [0,1]\}$, let $\theta_*:=\min \{\theta_g: g\in\mathcal G\}$.
 It then follows that $\theta_*>0$ and hence for any $0<\theta\leq\theta_*$, we have $\operatorname{Ric}_{g^\theta}>0$.
\end{proof}

\begin{remark}
An immediate consequence of the above is that, although $S^1\times S^3$ has a metric with $\sec^\perp>0$, it cannot satisfy $\sec^\theta>0$ for all $\theta>0$ with metrics that do not diverge (otherwise, it would have a metric with $\operatorname{Ric}>0$).\end{remark}

\subsection{Negative sectional curvatures}\label{subsec:negatives}
Although the first step in our deformation preserves $\sec\geq0$ from the product metric, the second step does not. In fact, for all $\theta>0$ there are planes $\sigma$ in $(S^2\times S^2,g^\theta)$ with $\sec_{g^\theta}(\sigma)<0$. This follows from an obstruction to positive first-order deformations observed by Strake~\cite[Sec. 4]{strake}. Namely, all zero planes in the Cheeger deformed metric $g=g_t$ from Section~\ref{sec:cheeger} are tangent to a totally geodesic flat torus, see M\"uter~\cite[Satz 4.26]{mueter}. Pick one such torus $i\colon T^2\hookrightarrow (S^2\times S^2,g)$, that intersects $\pm\Delta S^2$. The first-order deformation $g_s=g+s\,h$ on $S^2\times S^2$ induces a first-order deformation $i^*g_s$ on $T^2$. As observed by Strake~\cite[Lemma 4.1]{strake}, since $i(T^2)$ is totally geodesic, the first variation for the sectional curvature on $T^2$ coincides with the ambient variation:
\begin{equation}\label{eq:ddsinduced}
\frac{\mathrm{d}}{\mathrm{d} s}\sec_{i^*g_s}(\sigma)\Big|_{s=0}=\frac{\mathrm{d}}{\mathrm{d} s}\sec_{g_s}(\mathrm{d} i(\sigma))\Big|_{s=0}.
\end{equation}
In fact, this follows directly by differentiating the Gauss equation of $i(T^2)\subset (S^2\times S^2,g_s)$ at $s=0$. Let $i(p)$ be a point where $i(T^2)$ intersects $\pm\Delta S^2$. Then if $\sigma=T_pT^2$, $\mathrm{d} i(\sigma)$ is such that $\sec^\theta_g(\mathrm{d} i(\sigma))=0$, so the construction in Section~\ref{sec:firstorder} is such that \eqref{eq:ddsinduced} is positive. By the Gauss-Bonnet Theorem, $A(s)=\int_{T^2}\sec_{i^*g_s}\,\mathrm{vol}_{i^*g_s}=2\pi\chi(T^2)$ vanishes identically, so that
\begin{equation}
0=A'(0)=\int_{T^2} \left(\frac{\mathrm{d}}{\mathrm{d} s}\sec_{i^*g_s}\Big|_{s=0}\right) \,\mathrm{vol}_{i^*g}.
\end{equation}
Since the above integrand is positive at $i(p)\in\pm\Delta S^2$, it must also be negative somewhere. Together with \eqref{eq:ddsinduced} and the fact that $i(T^2)\subset(S^2\times S^2,g)$ is totally geodesic and flat, this means that $g_s$, $s>0$, must have some negative sectional curvature.

\subsection{Limiting case}
Since $\theta>0$ can be chosen arbitrarily small for our construction, a natural question is what happens to $g^\theta$ as $\theta\to 0$. By the above observations, the metric $g_s$ in \eqref{eq:gs} has some negative sectional curvature as soon as $s>0$. This implies that as $\theta\to 0$, the interval $0<s<s_\theta$ for which $g_s$ has $\sec^\theta>0$ shrinks until it disappears when $\theta=0$, since $s_\theta$ must also go to zero. In fact, if there was a uniform lower bound $0<s_*\leq s_\theta$ for all $\theta>0$, then the metrics $g_s$, $0<s<s_*$, would be such that the average sectional curvatures of any two distinct planes at the same point is positive, which in particular implies $\sec_{g_s}\geq0$, $0<s<s_*$, contradicting Subsection~\ref{subsec:negatives}. This is also reflected by the fact that the domain $K$ must be chosen compact in order for Proposition~\ref{prop:cpct} to hold, hence one cannot simply take $K$ to be the complement of the diagonal, see also Subsection~\ref{subsec:othercpcts}.

\subsection{Finite quotient}\label{subsec:rp2}
Our construction of metrics $g^\theta$ with $\sec^\theta>0$ on $S^2\times S^2$ can be made invariant under the antipodal action of $\mathds{Z}_2\oplus\mathds{Z}_2$, so that they induce metrics with $\sec^\theta>0$ on $\mathds{R} P^2 \times \mathds{R} P^2$. In particular, $\mathds{R} P^2 \times \mathds{R} P^2$ admits metrics with $\sec^\perp>0$.  Since such metrics come from a local isometric covering $(S^2\times S^2,g^\theta)\to\mathds{R} P^2\times \mathds{R} P^2$, they also do not have $\sec\geq0$ due to the above observations.

The first step in the construction gives rise to metrics invariant under $\mathds{Z}_2\oplus\mathds{Z}_2$, since it is a Cheeger deformation with respect to the $\mathsf{SO}(3)$-action, which commutes with the $\mathds{Z}_2\oplus\mathds{Z}_2$-action. As a side note, it was observed by M\"uter \cite[Satz 4.27]{mueter} that the induced metric on $\mathds{R} P^2\times \mathds{R} P^2$ at this stage is such that all its zero curvature planes are tangent to totally geodesic flat tori. The second and final step of our construction can also be made so that the resulting metrics are $\mathds{Z}_2\oplus\mathds{Z}_2$-invariant. Namely, this property is equivalent to the function $\phi\colon S^2\times S^2\to\mathds{R}$ in Proposition~\ref{prop:final} being $\mathds{Z}_2\oplus\mathds{Z}_2$-invariant, which can be achieved by defining the cutoff functions $\chi_\pm$ in a symmetric way.

\subsection{Biorthogonal pinching and isotropic curvature}
The biorthogonal curvature of a manifold $(M,g)$ is said to be \emph{(weakly) $\sfrac14$-pinched} if there exists a positive function $\delta$ such that $\tfrac\delta4\leq\sec^\perp_g(\sigma)\leq\delta$ for all $\sigma$. This notion can be extended to any dimensions by requiring that the average of any two mutually orthogonal planes is $\sfrac14$-pinched.
As observed by Seaman~\cite{seaman2}, this pinching condition implies that the manifold has nonnegative isotropic curvature. It was later proved by Seaman~\cite{seaman}, and independently by Micallef and Wang~\cite{mw}, that if an even dimensional compact orientable manifold $(M,g)$ with $b_2(M)\neq0$ has nonnegative isotropic curvature and positive biorthogonal curvature at one point, then $(M,g)$ is K\"ahler, $b_2(M)=1$ and $M$ is simply-connected. Consequently, our metrics of positive biorthogonal curvature on $S^2\times S^2$ cannot satisfy the biorthogonal $\sfrac14$-pinching condition, since $b_2(S^2\times S^2)=2$. Moreover, it also follows that such metrics do not have nonnegative isotropic curvature.

\subsection{Modified Yamabe invariant}
As observed by Costa~\cite{ezio}, the minimum of the biorthogonal curvature at each point is a \emph{modified scalar curvature}, with  corresponding \emph{modified Yamabe invariant} denoted by $Y^\perp_1(M)=\sup_g Y_1^\perp(M,g)$, where the supremum is taken over all metrics $g$ on $M$. It is observed that if a metric $g\in [g_0]$ is conformal to the standard product metric on $S^2\times S^2$, then $Y_1^\perp(S^2\times S^2,g)\leq0$. In particular, no metric conformal to $g_0$ can have positive biorthogonal curvature. However, as a direct consequence of the Theorem in the Introduction, we have that $Y^\perp_1(S^2\times S^2)>0$, see \cite[Thm 3 (1)]{ezio}.

\section{Other $4$-manifolds with positive biorthogonal curvature}\label{sec:4mflds}

In light of the above construction, it is natural to inquire how restrictive the positive biorthogonal curvature condition is on $4$-manifolds. As noted before, $\sec^\perp>0$ automatically implies $\operatorname{scal}>0$, however it does not necessarily guarantee $\operatorname{Ric}>0$ (cf. Subsection~\ref{subsec:ricci}). On the one hand, this means that $\sec^\perp>0$ imposes rather restrictive topological conditions on $4$-manifolds, e.g., vanishing of all the Seiberg-Witten invariants. On the other hand, such topological restrictions are by far not as strong as the ones implied by $\sec>0$, or even $\operatorname{Ric}>0$. For instance, $\sec^\perp>0$ does not guarantee finiteness of the fundamental group, as illustrated by $S^1\times S^3$ with the standard product metric.

This suggests that more subtle Synge-type obstructions should also not detect $\sec^\perp>0$. In fact, $\mathds{R} P^2 \times \mathds{R} P^2$ admits metrics with $\sec^\perp>0$, as discussed in Subsection~\ref{subsec:rp2}. Another relevant example in this context is the nontrivial $S^2$-bundle over $S^2$, which is diffeomorphic to the connected sum $\mathds{C} P^2\#\overline{\mathds{C} P}^2$, where $\overline{\mathds{C} P}^2$ denotes the manifold $\mathds{C} P^2$ with the opposite orientation from the one induced by its complex structure. We conclude by showing that this manifold also has $\sec^\perp>0$, using arguments similar to the $S^2\times S^2$ case. It is important to observe that it is also currently unknown whether $\mathds{C} P^2\#\overline{\mathds{C} P}^2$ admits a metric with $\sec>0$.

\begin{proposition}\label{prop:cp2}
There manifold $\mathds{C} P^2\#\overline{\mathds{C} P}^2$ admits metrics with $\sec^\perp>0$.
\end{proposition}

\begin{proof}
Similarly to the $S^2\times S^2$ case, $\mathds{C} P^2\#\overline{\mathds{C} P}^2$ admits cohomogeneity one metrics with $\sec\geq0$ invariant under the action of $\mathsf{SU}(2)$. In order to describe this initial metric, notice that the normal bundle of the usual embedding $\mathds{C} P^1\hookrightarrow\mathds{C} P^2$ can be identified with the vector bundle $(S^3\times\mathds{R}^2)/S^1$ over $\mathds{C} P^1=S^3/S^1$. Take two copies of the disk bundles given as tubular neighborhoods of the zero section of this vector bundle. Each one of them is the complement of a metric ball on $\mathds{C} P^2$, that is deleted to carry out the connected sum. It is then easy to see that
\begin{equation}
\mathds{C} P^2\#\overline{\mathds{C} P}^2=(S^3\times S^2)/S^1,
\end{equation}
by gluing along the boundary these two disk bundles. Here, the $S^1$-action on $S^3\times S^2$ is a product action, on $S^3$ via the Hopf action and on $S^2$ by rotation. The standard product metric on $S^3\times S^2$ then induces a submersion metric\footnote{We remark that this construction is very similar to the original gluing construction of nonnegatively curved metrics on the connected sum of two compact rank one symmetric spaces, which is due to Cheeger~\cite{cheeger} and was later greatly generalized by Grove and Ziller \cite{gzannals}. The only subtle difference is that $(\mathds{C} P^2\#\overline{\mathds{C} P}^2,g_0)$ has only one orbit that is a totally geodesic hypersurface (the boundary of the disk bundles glued together), while in the gluing construction the metric locally splits as a product near this hypersurface.}
 $g_0$ with nonnegative curvature on $\mathds{C} P^2\#\overline{\mathds{C} P}^2$. The cohomogeneity one action of $\mathsf{SU}(2)$ comes from the left-translation action of $\mathsf{SU}(2)=S^3$ on the first factor of $S^3\times S^2$, which induces an action on the quotient since it commutes with the above circle action.

Both singular orbits of this cohomogeneity one action on $(\mathds{C} P^2\#\overline{\mathds{C} P}^2,g_0)$ are $2$-spheres, that correspond to the zero section of the disk bundles that were glued together. The zero curvature planes are images via the submersion of mixed planes on $S^3\times S^2$ that are spanned by vectors orthogonal to the circle action field, cf. M\"uter~\cite[Satz 4.29]{mueter}. Thus, there is a circle's worth of zero curvature planes at every point, but any such planes tangent to a regular point \emph{must intersect}. At singular points, there are zero curvature planes orthogonal to each other, but all of them are not tangent to the singular orbit. This scenario is totally analogous to the Cheeger deformed metrics on $S^2\times S^2$, i.e., metrics obtained after the first step of our deformation (Proposition \ref{prop:geomcheeger}). More precisely, $\sec_{g_0}\geq0$, and $\sec^\perp_{g_0}>0$ on all regular points. Since the only points with zero biorthogonal curvature are along the singular orbits and all zero curvature planes are not tangent to these orbits, a first-order local conformal deformation using squared distance functions to the singular orbits, totally analogous to the one in Proposition~\ref{prop:final}, gives the desired metrics with $\sec^\perp>0$ on $\mathds{C} P^2\#\overline{\mathds{C} P}^2$ as a consequence of Proposition~\ref{prop:cpct}.
\end{proof}

\begin{remark}
As shown above, in order to construct metrics with $\sec^\perp>0$ on $\mathds{C} P^2\#\overline{\mathds{C} P}^2$, one can skip the first step in the construction for $S^2\times S^2$. This is an important observation, because differently from $S^2\times S^2$ with the standard product metric, the Cheeger deformation of $(\mathds{C} P^2\#\overline{\mathds{C} P}^2,g_0)$ with respect to the $\mathsf{SU}(2)$-action does not destroy \emph{any} zero curvature planes, see M\"uter~\cite[Satz 4.29]{mueter}.
\end{remark}

\begin{remark}
Since there is a circle's worth of zero curvature planes at every point on $(\mathds{C} P^2\#\overline{\mathds{C} P}^2,g_0)$, although the first-order local conformal deformation produces $\sec^\perp>0$, it cannot be used to produce metrics with $\sec^\theta>0$ for every $\theta>0$.
\end{remark}

redpajama_set_name: RedPajamaArXiv


text:
require "spec_helper"

RSpec.describe SubjectDataMailerWorker do
  let(:project) { create(:project) }
  let(:url) { "https://www.zooniverse.org/lab/123" }

  it 'should deliver the mail' do
    expect{ subject.perform(project.id, "project", url, ["zach@zooniverse.org"]) }.to change{ ActionMailer::Base.deliveries.count }.by(1)
  end

  context 'when there are no recipients' do
    it 'does not call the mailer' do
      expect(SubjectDataMailer).to receive(:subject_data).never
      subject.perform(project.id, "project", url, [])
    end
  end
end


text:
Barnes & Noble Implies Coronavirus Layoffs Are Inevitable
"This is a devastating situation in which to find ourselves," the company's CEO wrote to staff in a letter obtained by VICE.
by Alex Norcia
National book chain Barnes & Noble preemptively disclosed its plan for a forthcoming layoff strategy to employees on Tuesday, ahead of what seem like inevitable store closures due to the national economic chaos spurred by the global coronavirus pandemic.
"With the closure of stores, we are obliged to make the hardest of choices," CEO James Daunt wrote in a note to staff and obtained by VICE. "The truth is that we cannot close our doors and continue to pay our employees in the manner of Apple, Nike, Patagonia and REI. They can do this because they have the resources necessary; we, and most retailers of our sort, do not."
Daunt told his staff that, should store locations have to close their doors, staffers will "first make use of their Paid Time Off." After that, employees with a year or more of service will receive "up to" two weeks of pay. "Temporarily, and with sincere regret, on closure we lay off all those employees impacted with less than 6 months employment on the day of closure," he added. (Barnes & Noble did not immediately respond for comment.)
Are you a retail worker who is struggling amid the coronavirus pandemic? We want to hear from you. From a non-work device, email Alex Norcia at alex.norcia@protonmail.com or message him on Signal at 201-429-7024.
"This is a devastating situation in which to find ourselves and we understand the personal impacts of such action," Daunt wrote. "When a closed store is permitted to reopen, we will do so, and we intend to rehire."
Companies of all sizes have been scrambling over the last week as the coronavirus pandemic has led to shutdowns and a sudden unwillingness to enter brick-and-mortar stores like Barnes & Noble. Last Friday, Daunt predicted the crisis would cause an "unprecedented" decline in sales at the company, noting that it would likely have to "cut costs" in order to "weather this storm."
"No one knows by how much our sales will decline, nor for how long," Daunt wrote in a separate note to staff on Friday. "We do know, however, that the drop will be unprecedented and that we must assume this will be measured over a period of many weeks, and possibly of months."
Barnes & Noble, which has struggled ever since the rise of online booksellers like Amazon, hired Daunt in 2019 from the British bookstore chain Waterstones to turn things around. For the past two years, he has set out to make the stores smaller and more intimate—much like independent bookstores, where people could gather and browse.
Lauren Kaori Gurley contributed reporting.
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Tagged:amazonBooksbookstoreretail sales


The Equal Education Law Centre, Open Society Initiative for Southern Africa, and Open Society Foundation for South Africa will hold the Southern Africa Regional Consultation on the Human Rights Guiding Principles on the obligations of States with regards to the delivery of education by public and private schools, in Johannesburg from 14-17 August 2017. Joining the consultation will be 36 education and human rights stakeholders representing 12 countries in the region.
The last two decades have seen a significant increase in the scale and scope of non-state actors in education, particularly in developing countries. A critical question in each context is whether the involvement of the private sector in education is acceptable or not. What is still missing, however, is a broadly accepted understanding of the normative framework against which to make this assessment. While international human rights law has the potential to fill in this gap, the existing legal framework still needs to be clarified and consolidated.To do so, over the past 12 months various education stakeholders have been working together to develop a set of Guiding Principles that compile together existing customary and conventional human rights law as it relates to private actors in education. They are intended to be operational in and adaptable to different contexts, and to provide a basis for advocacy, policy development, and litigation.
As part of a broad consultative process to develop the Guiding Principles, various regional and thematic consultations are being convened over the course of 2017 with a range of stakeholders including civil society organisations, State representatives, human rights organisations, academics, education lawyers, international and regional organisations and other actors. Previous consultations have taken place in Bangkok for the Asia-Pacific region, Nairobi for East Africa and Paris for Northern America and Western Europe.
During this consultation, participants will learn more about the Guiding Principles and input directly into their development. Participants will bring their own context, experiences and challenges to the table in shaping a common understanding of the impact of private actors in education.


text:
GE Renews Customized Service Agreement with Royal Canadian Navy for LM2500 Gas Turbines
GE Marine announced it has renewed a long-term, customized service agreement (CSA) with the Royal Canadian Navy (RCN). The CSA covers an operating fleet of 24 GE LM2500 aero-derivative marine gas turbines plus spare engines used to power Halifax class frigates, GE announced.
According to Kris Shepherd, Vice President, General Manager, GE Marine, Evendale, Ohio, "GE has been providing the RCN with 20 years of customized service for their LM2500 engines, yielding significant performance and operational advantages in terms of improved reliability and the high availability of the gas turbines to power their surface combatants."
"GE builds on its relationship of trust and reliability with the RCN, dating back to the early 1990s. The CSA provides the Navy with high availability of the RCN's LM2500 fleet over the initial five years of contract support, as well as helping to promote supplier and skills development in Canada. The five-year contract comes with renewal and sustainment options until the eventual decommissioning of the Halifax class frigates," Shepherd added.
Other benefits of this CSA include formal and on-the-job training with GE and Navy personnel working side by side to maintain the LM2500 fleet, and assistance with procurement, inspection, technical support and materials inventory management.
The Royal Canadian Navy Halifax-Class Frigate Hmcs Calgary (Ffh-335) (U.s. Navy Photo)
Backed by GE's extensive network of global field service technicians, the RCN has access to GE services located throughout the globe, providing immediate onsite technical support 24-hours-a-day, 7-days-a-week. Similar to the original contract, the scope of the renewed contract includes:
Repair, overhaul and engineering support
Parts warehousing and inventory management (including spare engines, supply of spare parts and replenishment of inventory)
Field service representative support (home port and deployed)
Support of naval engineering school training curriculum for on-engine and equipment maintenance
Operational level maintenance
Supply and distribution of technical manuals
Customized agreements provide direct access to GE's global inventory of parts and spare engines, and the ability to tap into GE's worldwide service and support expertise such as training, maintenance, repair and overhaul services — all on an as-needed basis. With a CSA, navies can realize the full potential for their critical propulsion gas turbines while balancing performance and risk, along with predictable costs and less administrative oversight.
Check out Naval Library App to find out the specifications of the Halifax-class frigates.
Multistage WordPress Redirect Kit
Royal Navy OPVs leave Portsmouth for Indo-Pacific deployment


88: Yvonne has earned 88 Hero Points by being an active WPMU DEV member. They can use these to buy real stuff!
173: Yvonne has earned 173 Reputation Points by getting involved in the WPMU DEV community.
Thank you Vaughan, it now seems to be working. I thought since the WP site lives in a sub-directory that I would need to register the site using that subdirectory as opposed to the root domain but I guess not. Thanks for fixing!
Yvonne commented on Blog comments - good or evil?
Support has been enabled on the staging site. Not sure about the server logs, I will look into it.
Hi Nithin, I just tested all the plugins on the staging server copy of the site. I deleted each and every plugin except Snapshot and WPMUDEV Dashboard. I even deleted Akismet. After each plugin I deleted, I tried a snapshot which includes the plugins option. Even after deleting ALL plugins (except Snapshot and WPMUDEV Dahsboard), it still failed. I also tried a managed backup which also failed. Now what? If all plugins are deleted and it's still failing, that would seem to suggest there's no conflict with any of the plugins. I have also reset the secret key on the live site if you need to access that. Just please remember that it is a LIVE site and plugins cannot be deactivated on that site (landoverlandings.com) like I have been doing on the staging server.
there was a php file extension in the file name interfering with allowed file types.


Are you worried that someone close to you is struggling with an alcohol addiction? A dependence on alcohol gradually builds, so it takes time for the problem to develop. What alcoholism symptoms should you look out for? If you notice them, how can you help a loved one?
Someone with alcohol addiction increases their tolerance over the years. Meaning, they'll likely get to the point where they can't function without drinking.
Showing physical signs of alcohol addiction like severe weight loss or gain, flushed appearance, brittle fingernails, premature aging, and yellowing of the skin.
You might feel helpless if you recognize these signs in a friend or family member. You probably want to help, but they deny there's a problem (which is another common sign of alcoholism).
Therefore, it's extremely important to encourage them to seek treatment before it's too late. Let them know you're only doing this because you care. When you neglect to get help for addiction, alcoholism can lead to severe health issues or even death.
Additionally, you may have to hold an intervention. During these sessions, all loved ones have the opportunity to participate and discuss their concerns. Once the person sees how their disease impacts the people around them, they're more likely to decide to seek help.
If you or a loved one desperately wants to start a new, sober life, San Antonio Recovery Center can help. Our addiction treatment center provides a safe, comfortable environment where clients can begin their journey to sobriety.
At our facility, we use a 12-step philosophy and our treatment model works to heal you physically, emotionally, and spiritually.
Is addiction controlling you? You can overcome your addiction and regain control of your life. Our affordable rehab facility is ready to get you on the path to recovery. Contact us today at 866-957-7885.


text:
Call Me Today
A Legacy of Helping Families with Quality Representation
Franklin Post-Divorce Lawyers
Call Crosier & Hudson Family Law at
Unfortunately, a divorce does not necessarily mean the end of your family disputes. In fact, many problems continue to linger even after couples go their separate ways, especially if children are involved. While many issues are simply unforeseeable, it is always a good idea to work with seasoned Franklin divorce attorney in order to avoid common post-divorce disputes. No matter what type of dispute you and your former spouse is facing, our team at Crosier & Hudson Family Law is ready to help you work through your issues and reach cost-effective, lasting solutions.
Able to help clients resolve a wide-range of post-divorce matters, including:
Alimony modification
Child support modification
Mediation for divorce
Permanent parenting plans
Visitation rights
Discuss the details of your dispute during a private case review.
Fortunately, the law allows the opportunity for individuals to modify their divorce orders. This can only be done if the individual experiences a change in circumstances, and this request must be considered by the Court.
Valid reasons to modify a decree after a divorce include:
Dramatic change in income
Medical/Addiction issues
Safety of the child
Modifications can be complex, so having trusted family law attorneys work with you to prepare an effective modification request.
Caring Guidance, Experience Counsel
One of the distinguishing qualities of our firm is the ability to provide personalized counsel to each client. We understand the value of maintaining open communications and fostering close attorney-client relationships. This is how we can ensure clients are happy with the solutions we provide.
You can reach our office by calling (615) 645-1388.
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Meet Attorney Diane Crosier
Serving the people of Tennessee since 1992, Diane is known for compassionate, skilled representation. She is dedicated to excellence in her field and has undergone additional high-level training to earn and maintain her listing as a Rule 31 Mediator. This means she is well-versed in alternative dispute resolution, helping families stay out of court and amicably resolve their issues.
Diane is also backed by years of trial-tested counsel. If your case does move to court, she is prepared to fully protect your interests. Her reputation has led to her being voted into the "10 Best Family Lawyers" by the American Institute of Family Law Attorneys. This is just one testament to her commitment to client satisfaction, her sterling reputation in the legal community, and her industry leading insight.
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Meet Attorney Jill Hudson
Jill is a family law advocate, seeking to resolve family law concerns with an aggressive approach and a compassionate heart. After earning her J.D. from the Nashville School of Law, Jill embarked on her impressive legal career in fighting on behalf of those who need their voice to be heard by the legal system.
Not only has Jill previously worked as Deputy Director for the Tennessee Association for Justice, as well as the Director of Communications at the Tennessee Department of Mental Health, she was also selected to Super Lawyers® Rising Stars™ each year from 2017 to 2022. Her track record of success and advocacy in an effort to fight for the justice her clients deserve are a source of comfort for those she represents. With Jill on your side, you can rest assured you are in the hands of a Franklin family lawyer who will construct a highly-personalized course of action for your particular case.
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Crosier Hudson Family Law - Franklin Family Law Attorney
2000 Meridian Blvd. Ste 200, Franklin, TN 37067 View Map
Website: https://www.crosierhudson.com


Opel has released details of the 2014 Opel Insignia OPC Sports Tourer, the ultimate top-of-the-line performer with refreshed look and modified chasis, ahead of its public debut at the Frankfurt Motor Show in September.
The interior welcomes the drivers with pure motorsport ambience. The OPC gearshift knob, a completely new main instrument panel, a new performance steering wheel and center console exude premium flair.
With six-speed manual transmission and all-wheel drive, the Insignia OPC sprints from zero to 100 km/h in 6.0 seconds (Sports Tourer: 6.3 seconds) and reaches a top speed of 250 km/h (electronically limited).


On February 16, 2009 I attended the Vittadini show. They just sent me two pictures of my favorite items in their collection. I want you, my loyal readers, to see these beauties.
Vittadini did the smart thing. Not only was the collection shown on stationary mannequins so we could actually get up close and personal to the goods, but the venue was ideal. It was held in the International Center of Photography just up the block from the Bryant Park tents. In between bites of some of the best brownies washed down with sparkling water, I checked out the clothes and fell in love with Vittadini's sophisticated knitwear pieces, such as the forest green double knit cocoon coat over a dolman tunic and vicuna color lambskin skirt. The warm soft grey knit tunic detailed with band cut ribbon trim and matching pants, Also gorgeous, the midnight green panne velvet graphic print shift dress. Affordable, stylish and definitely on the money.
Vittadini - stylish and confident. No high-speed stumbling models. They just gave everyone all the time necessary to see beautiful, well made garments!


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About GS Construction
GS Management Team
Civil Pipeline
Infrastructure Rehabilitation
Terran Sandwith
Terran learned the construction business literally from the ground up by apprenticing under his father, GS founder Glen Sandwith. He started in the field as a project manager and estimator, while learning the roles and responsibility of every job at the company.
Now as President & General Manager, Terran is responsible for the overall direction and management of GS Construction. He charts the course, leads by example and ensures that GS Construction's Vision, Mission, and Values are upheld every day.
Terran is a graduate of York University's Schulich School of Business, and earned an Executive MBA from Queen's University.
Mike Durdle
VP, Municipal & Development; Operations Manager
Mike has worked his way from the bottom to the top at GS Construction. He started as a labourer in 2004, moving to Topman, Junior Foreman, Foreman, Superintendent, and then Project Manager. Now Mike oversees all our projects (except for our EPCOR work) as well as the overall operations of GS.
In addition to project oversight, Mike manages our equipment and truck fleet, and ensures that all our personnel are upholding the values and standards of GS Construction.
VP, Estimating; Senior Project Manager
Mark has progressed his way through the ranks into the GS management team. He started with us in 2000 as a Junior Foreman and has steadily worked his way up to head GS's Estimating department.
Mark is responsible for managing all systems involving lead generation, project tender procurement and pricing, and pre-qualification submissions. He also acts as Senior Project Manager on all GS projects.
Mark is a graduate of Northern Alberta Institute of Technology's Civil Engineering Technology program.
VP, EPCOR Division; Project Manager
Ryan came to GS Construction in 2012 after the company won a long-term renewable contract to replace infrastructure for EPCOR Utilities in the City of Edmonton. Ryan built and managed all aspects of GS Construction's new EPCOR Water Division.
Today, this core business unit successfully replaces approximately five kilometres of watermain each year. Prior to GS, Ryan had a successful career in project and contract management at Associated Engineering.
Ryan holds an Environmental Engineering Degree from Norwich University in Vermont, where he played for their Division III hockey program.
Laying the groundwork for your community since 1974
Specializing in Water & Wastewater Infrastructure Construction, we offer a full suite of services for Municipal & Civil Construction including rehabilitation & maintenance, new construction, land development, general contracting, and construction management.
© Copyright 2019 GS Construction | BubbleUP, an Edmonton Marketing Agency


The Tannoy CMS6 TDCB is a premium ceiling monitor system based around the Tannoy 165mm (6.5") point source, constant directivity Dual Concentric transducer. The loudspeaker is designed for situations where ceiling void space is limited, but where reference level music reproduction and high quality speech reinforcement are not to be compromised. The CMS6 TDCB is designed for blind mount applications, offering simple and cost-effective installation in new and existing buildings for applications where optimum audio performance is essential.
The drive unit of the CMS6 TDCB is the same drive unit used in comparable Tannoy reference-quality studio monitors. It incorporates the latest generation design features including a dual magnet assembly, HF Tulip WaveGuide and an injection molded polypropylene LF cone. The high frequency (HF) and low frequency (LF) sources are coincidentally aligned to a point source. The result is a smooth, uniform frequency response over a wide area of coverage unachievable with discrete, two-way designs in the vertical plane where massive cancellations appear off axis at the crossover frequency.
The CMS6 TDCB is designed for blind mount applications, offering simple and costeffective installation in new and existing buildings for critical applications.
of the CMS6 TDCB in conjunction with the THP 60 Watt transformer produces a high sensitivity, low loss system with wide dynamic range.
The CMS6 TDCB is available as the CMS6 TDCB-8 for use in 8 ohm operation and as the CMS6 TDCB-60, supplied with a THP 60 Watt transformer.


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Renaissance Art in Global Context
Renaissance Art in Global Context (AHIS30022)
Undergraduate level 3Points: 12.5On Campus (Parkville)
This class focuses on artistic encounters between European cultures and other world cultures between about 1300 and 1650. The goal is to place the art and culture of Europe into an expanded geopolitical sphere, and to explore particularly important and representative moments within the larger exchange of people, objects, ideas, and materials in the early-modern period. Readings will be drawn from both period sources and contemporary art history; and tutorials will focus whenever possible on objects in University and Melbourne collections.
In this subject students will:
Become familiar with major issues and objects in the study of cross-cultural artistic exchange in art history as a discipline and within the early-modern period
Learn to give feedback on others' ideas and expression through the workshopping of written work
Examine and evaluate how contact between artists and cultures created new and hybrid art forms, and develop a conceptual framework for analysing those forms
Understand how materials and technologies of art were affected by travel and trade
Gain a deeper understanding of basic shared themes in early-modern art, including a changing relation to the past in China and Italy, a search for new world audiences and new technologies in Benin and New Spain, or the engagement with violence and eroticism in Japan and France.
Students who successfully complete this subject will be able to:
Analyse and evaluate a variety of texts, images, objects, and materials
Participate in discussion and group activities and increase their creative and critical skills through workshopping and collaboration
Independently devise and articulate research and ideas in both verbal and written modes.


Residents of Kitagata subcounty in Sheema district have asked the Government to intervene and save the Kitagata Hot Spring from degradation.
They said the spring might be extinct if the activities threatening it are not stopped. Residents said construction activities on the Ishaka- Kitagata-Kagamba road have greatly affected the hot spring and accused Dott Services, the company working on the road, of not doing enough to protect it from being degraded.
Nazario Bishanga, the overseer of the hot spring, said Keitajonjo wetland water system was diverted, resulting in the flooding of the spring.
"They diverted the water system, including rivers, but did nothing to protect the spring from flooding," he said.
The residents have resorted to collecting money from visitors and tourists to buy murram, which they pack in sacks and put around the hot spring to block water from entering it.
"Dott Services are selling to us the murram, yet they are the ones supposed to protect the spring," Bishanga said.
However, the Dott Services public relations officer, Emmanuel Nowayesu, said they diverted the river to allow construction of a bridge.
He said the river water was diverted to another channel, but not the hot spring, adding that it is what was indicated in the environmental impact assessment.


Ryan Nelson got his first on this night full of firsts!! More Cordray's First Deer Hall of Fame members. I've got to order more free shirts!
Turner Hill got his first deer tonight, too. One more in the Cordray's First Deer Hall of Fame!
Hunter Hardwick got his first deer tonight! He joined the Hall of Fame, too!
Travis Easler got a buck with 11 points and 2 kickers. It had a 21 inch outside measurement and 19.5 inch inside.
Max and Lee Sparwasser show off Lee's big buck! She didn't want to do it, but she took the plunge to join the Hall of Fame!
Bert Sorin is the leader of the BUCK OF THE WEEK so far... 130 lbs. dressed weight!
Bert Sorin got an 8 point that weighed 194 lbs. live weight.
Austin Lakes got his first deer Saturday morning!
Summer Williams got a 10 point buck for her first!
Alex Shaw got his first deer and his entrance into the Cordray's First Deer Hall of Fame!
Donna Lewis got an 11 point!
Gus Covington's big buck is the leader in the BUCK OF THE WEEK! Dressed weight 103.8 lbs.
Jonathan Millsaps got a doe as his first deer. He joined the Cordray's First Deer Hall of Fame!
Reese Havens brought in a big buck tonight!
Durst Payne got this unusual 6 point buck in during the day today!
Peter Plott's buck weighed 95.4 lbs. dressed weight.
Ronnie Burbage got a nice buck!
Introducing a new product! Onion burger!
We've been field testing it for a few weeks and have gotten rave reviews. The price will be 50 cents more per lb. with a 10 lb. minimum. Give it a try on your next deer. We think you'll love it!
John Sparkman hit the jackpot! His first deer ever got him the usual Hall of Fame treatment with a free hat or shirt from Cordray's. He also had the biggest buck last week with a dressed weight of 113. 6 lbs. So he also goes home with $50 worth of Cordray Farms Beef!
Glenn Brown got this big 9 point!
The big bucks are beginning to get stupid! Sunday night Kenneth got 5 shoulder mounts in the 4 hours we were open! All were nice mountable bucks.
Almost all the bucks are showing clear signs of being in rut. We're hearing reports of sightings of plenty of lone bucks chasing does. The rain and wind has knocked most of the acorns from the trees so stands near oaks are in high demand right now. Some folks say that the full moon will bring the rest of the does into heat, so all the hunting should pick up even more toward the weekend. Let us know what you're seeing and we'll pass it on!
Gus Covington's buck was an 8 point weighing 174 lbs.
John Sparkman got a big one for his first and got the Cordray's First Deer Hall of Fame treatment with the help of Daniel Rogers!
Buck was a 7 point and weighed 185 lbs.
Diane Oliver took a big wildhog!