The stranger, the mother and the Algerian revolution

In The Myth of Sisyphus Albert Camus seeks his own answer to the question that Kierkegaard, Dostoevsky and Nietzsche have bequeathed to us: is it possible to live without God, without any hope of salvation as death looms? At first sight the outlook is bleak: we are, says Camus, thrust into existence as into an exile with no return. From the viewpoint of eternity, we are passions that can never be satisfied, cries that go unheard. Human beings and the world do not mix. He sums up man’s tragic existence in one word: the absurd. It emerges from the fracture between the soul’s desperate desire for clarity and the incomprehensibility of reality, the gap between our craving for meaning and the cold grave that awaits us. Camus understood early on that reason – the traditional weapon of philosophy – would always be powerless to address the futile, ill-fated game of existence.

Even so, the work essentially represents a manifesto for revolt and the ecstasy of life. It is a passionate revolt that answers the question of the meaning of life: if there is any raison d’étre worthy of the name, it is born of a piece of trembling flesh. Desire, affection and generosity: these are the bread and wine of the mortal human being. Anyone who does not live to serve an immortal Creator must become a creator himself. The absurd man, as Camus calls his hero, transforms his days into a work of art, a defiant celebration of the battle which, as he knows from the outset, must eventually end in his death. Camus quotes Nietzsche: “It is not a question of eternal life but of being eternally alive.”[1] It is the destiny of the absurd man to try to make something out of the nothingness, to grasp his life from death itself. Sisyphus shows us the way: happiness is the leap we make both with and in spite of our burden.

This insight is already there in the young Camus’ poetic meditations on the “home of the soul” which saw him come into the world in 1913: Algeria. It was here, in the generous sunshine of French Algeria, that the poet’s sensibility was formed. The 20-year-old’s reflections on the home country exude a sensual amor fati, a dream that the flesh might provide an escape from exile and conquer the absurd. Here he depicts the nuptial feast between people, the earth, the sun and the sea. His hometown, Algiers, writes Camus, worships the body, warmth and an eternal present. “For those who are young and alive everything in Algiers is a refuge and a pretext for continuous triumph: the bay, the sun, the red and white terraces facing the sea, flowers and sports grounds, the cool legs of young women.”[2] The Algerians’ simple request for happiness amidst their daily misery reveals man’s innocence with regard to life and death.

Perhaps this identification with the vibrant bodies on Algeria’s beaches expresses an instinctive rebellion against the serious case of TB that had, since his adolescence, pitched the author between hope and despair. The disease threatens to deprive him forever of the Algerian sun he loves so much, and which, bedridden, he could glimpse from his window in the simple working-class suburb of Belcourt. “In this land that offers so much life, everything exudes a terror of death,” he writes from his sick bed. Anyone who understands how to live without hope in the future sees the moment as his only kingdom.

Camus survived his illness and in the early 1940s left Algiers for metropolitan France and Paris. There he published in just one year, 1942, what he called in a diary entry “The triptych of absurdity”: the essay on The Myth of Sisyphus, the play Caligula, and the novel The Stranger. This would be the start of his big breakthrough. He was soon standing shoulder to shoulder with the monstres sacrés of French existentialism, Jean-Paul Sartre and Simone de Beauvoir. He was not yet 30 years old.

One of the issues the author examines in his essay on Sisyphus – “Knowing whether or not one can live without appeal is all that interests me,”[3] – gets its acid test in The Stranger, a literary portrayal of the obsessions of the French-Algerian Meursault. We accompany him from the days surrounding his mother’s death to the day before his execution. It is Meursault himself who tells the story, oscillating between insightful indifference and moments of sensual pleasure. He knows that life itself offers no compelling reason for either the one or the other; only chance, sensuality and spontaneous impulse are able to guide a life led without any higher meaning. A man can cry or not cry at his mother’s funeral, shoot or not shoot an Arab on the beach, marry or not marry a woman who declares her love for him. When all is said and done, everything is equal and people are essentially innocent when dealing with the absurd vicissitudes of life. “As if familiar paths traced in summer skies could lead as easily to prison as to the sleep of the innocent,” says Meursault.[4] In Camus’ preface to the US edition, the young Franco-Algerian is described as a martyr for the absurd. He is a man who refuses to cheat. Neither Church, State nor morality can persuade him to give up the truths of the heart. He is at once a Raskolnikov and a Josef K, but with the important difference that he never seeks to do penance. Meursault feel no remorse, nor does he try to convince anyone that he does. He does not speak unless he has something to say. Those who keep their thoughts to themselves are not swayed by public opinion.

Ultimately, it is Meursault’s indifference that leads to his downfall. According to the prosecutor, he “[buried] his mother with a crime in his heart”, and anyone who has killed his mother – morally speaking – is cut off from human society in the same way as someone who strikes his own father down with a killer’s hand.[5] If such a callous soul goes free, then an abyss opens up that can swallow society whole. The fate of Meursault depicts the anatomy of existential alienation through the image of a lucid, absurd man who is sentenced to the guillotine “in the name of the French people” in order to protect the national community against the most dangerous crime of all: patricide.

So much, then, for the narrative’s purely existential and universal human themes. But in the margins, a different story is playing out. It makes itself heard in a number of disturbing questions: Why does Meursault fire four more shots at an already lifeless body (“flooded with joy”, as Camus puts it in an earlier draft of the novel)?[6] Why are there no Arabs present at the trial? Why are so many of them in jail and why are they all nameless? And why is a Frenchman – who has just killed an Arab in Algeria – sentenced to death by the French colonial authorities for not weeping at his mother’s funeral? What kind of social order has Meursault struck out against?

The two stories play out on two different stages and portray two distinct types of alienation. In the first one, Meursault appears as an isolated Everyman: the dismal vicissitudes of fortune lead the young stranger to act in a certain way (although it could just as easily have been another one); to kill a certain person (although it could just as easily have been another one); and to be sentenced to death by a certain fatherland (although it could just as easily have been an another one – “why not German or Chinese?” as Meursault wonders). He is the sun’s hapless puppet – it is the “murderous sun”, “the whole beach, throbbing in the sun” that drives him towards the Arab – and when the trial judge asks him to explain his motives, Meursault replies that it was “because of the sun”.[7]

The historical reality of Algeria bursts forth onto the second stage and the anonymous figure of the dead body now reveals a concrete political stranger: the Arabs of colonial Algeria. This people, reduced to a kind of natural backdrop for the European Algerians’ apparently carefree existence, show by their mere presence that something is out of joint in French Algeria. If Meursault was the absurd hero on the first stage, here he appears instead as a typical French settler whose power is underpinned by physical violence (he is a Frenchman who kills an Arab), military superiority (the revolver defeats the knife), and the discriminatory legal system (focusing on the deceased French mother rather than on the murdered Arab). Meursault’s life choices emerge, despite his indifference, as part and parcel of a specific colonial reality. He is a pied noir, a French colonist in l’Algérie Française, who says he followed his “natural feelings” on the day he killed an Arab. Where the stranger of absurd existence sees only a series of disconnected moments, with no internal connections or any deeper meaning, the political stranger uncovers an underlying solid structure, a discriminatory machinery that ultimately rests on murder.

In other words, the colonial regime’s inherent symptom (political alienation) permeates the philosophical narrative (the sense of alienation before the absurd) as its internal negating movement. Camus’ novel hints, either consciously or unconsciously, at the deep divisions that threaten to blow up l’Algérie Française from within – and force the author to consider the circumstances of his own historical and political existence. History’s dark shadow falls over the story: between European Algerians and Arabs there is no possible reconciliation, no communication, only a fight to the death. The wordless meeting not only signals that they are strangers to each other, but also that Algeria is not big enough for both of them.

When, in the posthumously published autobiographical novel, The First Man, Camus looks back on his childhood, he often returns to the atmosphere that prevailed in Algeria at the time when the novel was written. We lived, he says, cheek by jowl with this “so close and yet so foreign people”. With their “hard, impenetrable faces” and sheer numbers, the Arabs constituted “an invisible threat that we could feel in the air”.[8] The natives’ stares and muffled murmurs reminded European Algerians that they were temporary strangers in the country and that they had made themselves masters at the cost of Arab blood. It would not be long before Algeria’s population would be forced to leave the beaches only to bathe in each other’s blood.

Early May 1945 saw the first steps towards open warfare. In the midst of the French celebrations of the defeat of Nazi Germany, Algerian nationalists took the opportunity to remind them that Algeria also had the right to liberty and independence. They defied the ban on symbols of Algerian independence and refused to give way to police threats. Bloody confrontations followed: after some 100 European civilians had been found dead, the French embarked on a senseless massacre. Tens of thousands of Arabs were killed and the prisons were filled with suspected nationalists. Camus took up his pen and wrote a number of articles demanding justice for the Arab people. He argued that radical reforms were needed to reduce the disparities between European and native Algerians. No civilization could survive when “dignity” had been sidelined for a large proportion of the population. The French had to realize that the country’s Arab and Berber population were a “sister people” and that together they should lay the foundations for a “New Mediterranean Culture”. Otherwise, he warned, the whole country threatened to shatter from within; it was time for all good men of France to do their duty for the native population: “It is only through the infinite power of justice that we will be able to win back Algeria and its inhabitants.”[9]

For many Algerians, however, the massacres were a decisive turning point. Mistrust of France had grown to such a degree that that there was no turning back. Since, after over a century of French rule, Arabs were still not welcome as equal citizens, they now needed to create their own Algerian state. Inspired by the worldwide anti-colonial struggle, the Algerian Liberation Front (FLN) proclaimed a national revolution on 1 November 1954. They accused the French of betraying their own ideals: it was now the Algerian nationalists who operated in “la pure tradition de la France révolutionnaire.”[10] The Declaration of Human Rights and the right to self-determination must also apply to the Algerian people.

For eight years France would fight, to borrow a phrase from Simone de Beauvoir, “a war that dares not speak its name”.[11] The official line in France was that it was neither a revolution nor a war of independence, merely temporary disturbances instigated by fanatical terrorists. The French nation is one and indivisible, explained the regime, and is no more likely to leave Algeria than it would leave Provence or Brittany. One of the bloodiest wars of the post-war period begins and France stops at nothing to keep Algeria French: summary executions and use of prohibited weapons; systematic torture and terror against civilians; population displacement on a massive scale; and concentration camps. Nearly one million people lost their lives. During the last year of the war Algeria was a bloodbath and hundreds of thousands of Europeans were forced to flee their former homeland. A regular French civil war was soon looming large: French officers in Algeria threatened to overthrow the “traitor” De Gaulle for starting to negotiate with the Algerian nationalists. The export version of French military chauvinism fought back against the mainland government, but De Gaulle held his ground and in July 1962 Algeria won its independence.

“Right now I have I have pain in Algeria, as others have a pain in their lungs,” explains Camus shortly after the revolution broke out in 1954.[12] The increasingly savage war and the bitter debates in France forced Camus into a difficult position. He tried to develop his own stance in the ideological war that was taking shape in metropolitan France. On the one hand, he turned against the supporters of French Algeria who turned a blind eye to colonial racism and the excesses of the French army. On the other, he was unable to reconcile himself with the radical Left who embraced the Algerian nationalists unreservedly. While the French Right saw Camus as a traitor because of his criticism of French colonial policy, the Left attacked him for betraying the tradition of anti-fascist resistance. In Algeria, said Sartre, what we are seeing is the striptease of western humanism. France boasts of its universal values while it burns Muslims alive; but now the former Untermensch (“sous-hommes”) is rising up against us to assert his humanity.[13]

Perhaps the most difficult challenge for Camus came from those who saw the anti-colonial struggle as some kind of continuation of la Résistance, the French Resistance movement during the German occupation. Personally active in the Resistance, at the end of the war Camus praised those French people who, with weapons in hand, rose up against the oppressor: “A people that wants to live does not wait to be given its freedom. It takes it.”[14] The Algerian nationalists could have made these words theirs, but Camus consistently refused to equate Algerian nationalism with the patriotism of the French Resistance. For him there was no such thing as an “Algerian people”. He speaks consistently of “Arabs”, not “Algerians”. He therefore sees the nationalists’ demands for independence from France as an expression of political naivety, where they weren’t actually being manipulated by outside forces. In one of his last pieces on the war, Camus warns of the “new Arab imperialism” he feels is emerging and which he claims has its roots in Nasser and the Soviet Union. The FLN, says Camus, thus runs errands for other powers and is consequently associated with a violence that “no revolutionary movement has ever accepted”, not least the French Resistance.[15] Unlike his former friends on the radical Left, Camus has little faith in historical reason, revolutionary violence or the communist revolution: for him, not even Marx can offer people a way out of their eternal exile. In the Cold War, Camus is significantly closer to the West than the East.

Camus never breaks completely with the premise of French colonialism: that the most fervent wish of the colonized is to be recognized as equals in French civilization. The principles of the French Enlightenment – freedom, equality, fraternity – are interwoven with a colonial paternalism. Despite the torture, racism and the obvious betrayal of the native Algerians, Camus never stopped seeing France as “the best possible future for the Arab people”.[16]

At the same time, we cannot be totally sure where Camus really stood. He knew that everything he said could be used against those of his closest friends who still remained in Algeria. In the late 1950s, Camus said that he could no longer take part in the debate: “I will not, under any circumstances, allow any statement of mine to be used to give a clear conscience to a reckless fanatic who shoots into a crowd in Algiers, where my mother and all my nearest and dearest live.”[17] In January 1956, he made his last public appearance at a conference in Algiers, which, despite receiving death threats, he had helped to organize. Under the heading, “Appeal for a Civil Truce”, Camus calls for a clear declaration from all sides in the war to keep civilians out. Nothing, he says, can ever justify an innocent death.[18]

The theme of “the innocent” is already present in The Myth of Sisyphus and runs even deeper through the political philosophy that he developed under the influence of both World War II and the events in Algeria. In the novel The Plague, the play The Just Assassins, and the essay “The Rebel”, the innocent play a critical role in the distinction he establishes between revolt (which never allows civilian casualties in the struggle) and revolution (for which the end always justifies the means). The revolutionary dream of All or Nothing (absolute justice) must give way to a new social cogito as espoused by Camus in “The Rebel”: “I revolt, therefore we are”.[19] As with the people in The Plague, besieged by the arbitrary boils of death, all opposition must take its starting point in a desperate solidarity in the face of the gruesome aspects of our common destiny. We must not add to the injustices of the present in the name of justice tomorrow; that, too, bears the weight of absurdity on its shoulders.

It is in the light of these ideas that we should understand the famous yet enigmatic comment Camus made in his Nobel Prize acceptance speech in December 1957. A young militant Algerian confronted him and demanded an explanation of his stance towards the ongoing war:

I have been and remain an advocate of a fair Algeria, where both peoples can and must live in peace and equality. I have said and repeated that we must do what is right for the Algerian people and grant them a fully democratic government. […] I have always condemned terrorism. I must also condemn a terrorism that strikes blindly, in the streets of Algiers, for example, and which one day might strike my mother or my family. I believe in justice, but I will defend my mother before justice.[20]

Who then is this innocent mother – Catherine Sintes Camus? How are we to understand this mother figure that recurs throughout his writings? In a notebook from the 1930s, the 22-year-old Camus writes that, “my whole sensibility is based on my curious feelings for my mother”.[21] We have already seen that the mother figure appears in the story of Meursault, where the son’s apparent lack of concern at his mother’s death is a prelude to the moral machinery that ultimately condemns him to death. In the early autobiographical story, “Between Yes and No”, the son – disconsolate and guilt-ridden – beholds his silent mother and wonders whether he really loves her: “He feels sorry for his mother, but is it the same as loving her?”[22] In The Plague, Doctor Rieux receives a visit from his mother in Algeria and they “love one another in silence” without ever managing to “confess their affection” properly. In The Misunderstanding, however, written at the end of World War II, it is the mother who kills a man who later turns out to be her son. When Catherine Camus finally emerges in flesh and blood in his autobiography, her son is facing trial. It is she who makes him realize his limitations: “He who talks incessantly is incapable of saying in thousands of words what she can say in just one of her silences.” The memory of his early childhood and his practically deaf and dumb mother spending days on end sitting alone, quietly by herself, stuck in his memory like “glue in the soul”. She – the embodiment of innocence – fills her son with a sense of a debt unresolved.

Perhaps one might even say that the trial scene in The Stranger is a sort of literary dress rehearsal for the political predicament in which the quirks of history would place Camus: what sort of crime has the son actually committed? Is it the murder of the Arab or the betrayal of the mother? Is it justice or innocence? Even as the War of Independence was breaking out in Algeria, Camus wrote in his notebook: “Nothing in the world, even if it were innocent and just, could ever make me abandon my solidarity with my mother, who is the most important thing in the world to me.”[23]

***

Albert Camus urged his readers to look for “the dark and the instinctive” in his work, the points of concentration where literature, philosophy and life meet, the dimensions in which the universal is interwoven with both the historically specific and the individually unique. The literary characters in his writings – the mother, the Arab, justice, the sun, innocence, the absurd and revolt – reappear in new combinations throughout his life and works. During the war it was, as we have seen, the innocent mother who raised the barricade against the Arab revolution for justice. This is not the place to speculate further on the libidinal economy at work in these conceptual configurations and dislocations, but it may be worth at least asking about the place of Camus’ own father in this logic; the Lucien Camus whom he never got to meet because he sacrificed his life for the French nation in another war, World War I. It should be remembered that, indirectly, the question of the father already plays a decisive role in The Stranger – society must be safeguarded against patricide, “the most heinous of all crimes”, because it is the father who binds society together. One can wonder what it might mean for a son, in this case Camus himself, to lend his support to the Algerian nationalist attack on the very fatherland, la patrie, that his father sacrificed his life for? A father whose death also gave his surviving son the morally-privileged status of “pupille de la nation” (a war orphan, “adopted by the French nation”).


Cardiovascular complications are more prevalent in individuals with conditions associated with insulin resistance. Knowledge is evolving concerning the mechanistic link between insulin resistance and vascular disease, but complete clarity does not exist. This is a high-priority area of investigation because individual and societal costs associated with diabetic vascular complications are increasing, and the need to elucidate effective therapeutic targets and intervention strategies with minimal off-target side effects is urgent.

Insulin-resistant conditions such as type 2 diabetes and diet-induced obesity are associated with an altered endothelial cell phenotype, i.e., endothelial dysfunction. A crucial component of endothelial cell dysfunction is reduced nitric oxide (NO) bioavailability. Reduced NO bioavailability contributes importantly to pathologies specific to large (atherosclerosis, cardiomyopathy) and small (retinopathy, nephropathy, neuropathy) blood vessels ([@B1]). Oxidative stress, hyperglycemia, lipotoxicity, activation of the renin-angiotensin system, and increased proinflammatory cytokines are systemic disturbances in patients with conditions associated with insulin resistance. Each of these factors contributes independently and synergistically to decreasing NO bioavailability by impairing its synthesis and/or by enhancing its degradation ([@B2]). In this issue of *Diabetes*, Sukumar et al. ([@B3]) demonstrate that endothelial cell--specific insulin resistance increases NADPH oxidase (Nox) isoform 2 (Nox2) expression to an extent that elevates superoxide anion (O~2~·^−^) production and precipitates endothelial dysfunction.

Reactive oxygen species (ROS) are highly reactive metabolites of oxygen that participate in oxidation-reduction reactions ([@B4]). A traditional definition of oxidative stress is when ROS accumulation overwhelms the cellular antioxidant defense mechanisms and the redox state of the biological compartment is shifted toward one that is more oxidizing ([@B5],[@B6]). Cellular sources of ROS include NADPH-dependent oxidases, xanthine oxidase, lipoxygenases, mitochondrial oxidases, and NO synthases ([@B7]). Nox is a major source of O~2~·^−^ in insulin-resistant humans ([@B8],[@B9]) and mice ([@B10],[@B11]). Seven isoforms of Nox have been described in mammals ([Fig. 1](#F1){ref-type="fig"}). Each isoform comprises a core catalytic subunit, i.e., Nox1--Nox5 and dual oxidase (DUOX) 1 and DUOX 2 ([@B4]). Each Nox catalytic isoform contains up to five regulatory subunits that determine *1*) maturation and expression of Nox and DUOX subunits in biological membranes (e.g., p22phox), *2*) enzyme activation (e.g., p67phox, RAC1/RAC2), and *3*) spatial organization (e.g., p47phox). Endothelial cells express Nox1, Nox2 (gp91^phox^), Nox4, and Nox5; vascular smooth muscle cells express Nox1, Nox4, and Nox5; and adventitial fibroblasts express Nox2 and Nox4 ([@B5]).

![Hypothesized contributions from Nox2-generated oxidative stress to the process of atherosclerosis. When NADPH donates electrons to molecular oxygen via Nox2, O~2~·^−^ generation occurs. Oxidative stress occurs when O~2~·^−^ production exceeds the antioxidant capacity of the immediate environment. Oxidative stress can contribute to atherosclerosis progression by various mechanisms ([@B7]). ↑, increase; ↓, decrease. Mice with endothelial cell--specific insulin resistance (ESMIRO and IR^+/−^ mice) exhibit increased Nox2-mediated O~2~·^−^ generation to an extent that evokes endothelial dysfunction. Treatment of ESMIRO and IR^+/−^ mice with the Nox inhibitor gp91ds-tat (+gp91ds-tat) or endothelial cell--specific knockout of Nox2 in ESMIRO mice (ESMIRO / Nox2^y/−^) reduces O~2~·^−^ production and restores endothelial function. \*Endothelial function was assessed in the current study (3), but the influence of endothelial cell--specific Nox2 deletion on other potential contributors to atherosclerosis depicted in the figure has not been investigated. VSMC, vascular smooth muscle cell; MMPs, matrix metalloproteinases.](1818fig1){#F1}

Earlier work from this laboratory group indicates that mice with germ-line haploinsufficiency of the insulin receptor (IR^+/−^ mice) display hypertension and an age-dependent attenuation of arterial vasorelaxation that is associated with elevated Nox-mediated O~2~·^−^ production ([@B12],[@B13]). Importantly, impaired endothelial function in IR^+/−^ mice has physiological relevance in models of endothelial injury ([@B14]) and atherosclerotic plaque development ([@B15]). A valid concern when interpreting these results is that metabolic disruptions known to exist in IR^+/−^ and apoE mice might contribute to impaired endothelial function. To address this, mice with endothelium-targeted overexpression of a dominant-negative mutant human insulin receptor (endothelium-specific mutant insulin receptor overexpressing mice \[ESMIRO\]) were studied ([@B10]). Compared with wild-type littermates, ESMIRO mice have preserved glucose homeostasis and are normotensive. However, endothelium-dependent vasorelaxation and endothelial NO synthase (eNOS) phosphorylation are blunted in aortae from ESMIRO versus wild-type mice, despite similar eNOS protein expression. Because mRNA expression of Nox2 and Nox4 was increased in endothelial cells and aortae of ESMIRO mice, and endothelium-dependent dysfunction was normalized when aortae from ESMIRO mice were incubated with a superoxide dismutase mimetic, the authors concluded that Nox-mediated O~2~·^−^ generation contributed importantly to dysfunction observed in arteries from mutant mice ([@B10]).

Sukumar et al. investigated a relatively new approach to the challenge of preventing arterial dysfunction in the context of insulin resistance, i.e., to suppress O~2~·^−^ generation locally by disrupting an NADPH enzyme complex located in the endothelial cell. First, the authors confirmed the preserved metabolic phenotype and disrupted vascular phenotype that exists in IR^+/−^ and ESMIRO mice, and demonstrated that exaggerated O~2~·^−^ production and endothelial dysfunction exhibited in aortae from both groups could be attenuated by acute or chronic Nox inhibition using gp91ds-tat ([@B16]). Because Nox2 mRNA expression was elevated in endothelial cells from IR^+/−^ and ESMIRO mice and Nox2 siRNA suppressed O~2~·^−^ generation in endothelial cells from ESMIRO mice, the authors sought to determine whether Nox2--mediated O~2~·^−^ production is sufficient to contribute to arterial dysfunction in the context of insulin resistance. To explore this, mice with endothelial cell--specific insulin resistance and deletion of Nox2 were generated i.e., ESMIRO / Nox2^y/−^ mice. As hypothesized, increased O~2~·^−^ generation and vascular dysfunction were prevented in endothelial cells and/or aortae from ESMIRO/Nox2^y/−^ versus ESMIRO mice. Importantly, systemic glucose homeostasis and blood pressure were similar between groups. These findings provide strong evidence to consider the Nox2 isoform as a therapeutic target for the treatment or prevention of vascular disease in the context of insulin resistance.

Strengths of this particular study include *1*) the use of double transgenic mice wherein insulin resistance and Nox2 deletion is created specifically in the endothelial cell; *2*) demonstrating that systemic glucose homeostasis and arterial pressure was not altered in double transgenic versus ESMIRO mice, *3*) using redundant methodologies to confirm O~2~·^−^ generation; and *4*) verifying that Nox2 expression and O~2~·^−^ production in the context of insulin resistance was negated in denuded arteries. Further exploration of *1*) the systemic metabolic environment (with particular focus on factors known to alter Nox2 expression, e.g., angiotensin II), *2*) arterial function (e.g., acetylcholine-evoked vasorelaxation and receptor/nonreceptor-mediated contraction ± NOS inhibition ± endothelial denudation), *3*) the impact of Nox2 inhibition/deletion on the function of other tissues known to express high levels of Nox2 (e.g., pancreatic β-cells) ([@B17]), and *4*) indices of vascular signal transduction (e.g., in vivo and/or in vitro insulin-mediated arterial eNOS phosphorylation and NO production) in ESMIRO/Nox2^y/−^ and ESMIRO mice could yield interesting findings to provide even greater insight. Future research questions arising from this important contribution to the field include *1*) Does Nox2 deletion in endothelial cells influence the expression/function of other Nox isoforms throughout the arterial wall and/or unmask beneficial effects of vasoprotective (?) isoforms such as Nox4 ([@B5],[@B18],[@B19])? *2*) Will endothelial cell--specific deletion of Nox2 mitigate the endothelial dysfunction and hypertension that exist in the context of obesity in high-fat diet--fed mice ([@B11],[@B20]) and/or atherosclerosis in ApoE mice ([@B15])? *3*) Because females have lower Nox expression and activity than males ([@B21]), do sex-specific differences concerning Nox2-mediated O~2~·^−^ generation exist?

The prevalence of diagnosed and undiagnosed cases of type 2 diabetes---a condition characterized by insulin resistance and endothelial dysfunction---is estimated to increase to 33% by 2050 ([@B22]). Sukumar et al. have provided novel mechanistic insight into defining a molecular target that should be pursued further in an effort to delay the onset and/or lessen the severity of vascular disease that exists in patients with conditions associated with insulin resistance.

See accompanying brief report, p. 2130.

J.D.S. is supported by National Institutes of Health grant 2R15HL091493, American Diabetes Association Research grants 1-12-BS-208 and 7-08-RA-164, and the University of Utah College of Health and School of Medicine.

No potential conflicts of interest relevant to this article were reported.

The author is grateful to Anthony J. Donato (Department of Internal Medicine, University of Utah) and Sihem Boudina (Department of Molecular Medicine, University of Utah) for their expert advice and critical evaluation of the manuscript. The author also thanks Paula Fernandes D'Elia and Matthew Walker (University of Utah) for their help with the figure.


Duke is where Tweedy himself went to medical school, and he told me that although most of his classmates there were from colleges more selective than U.M.B.C. and families with more money than his, “I scored in the top 20 percent of my class during that first year of basic science classes, which are the toughest part of med school.” U.M.B.C. had prepared him well, not just academically but also, he said, by making him feel that he was “part of something more than just your individual attainment.”

He recalled that before he decided to go there, several Ivy League colleges tried to recruit him — for basketball. In contrast, he first came on to U.M.B.C.’s radar because of his aptitude for science.

More than one in four U.M.B.C. undergraduates qualifies for federal Pell grants, meant to serve low-income families. About 45 percent are white, while 18 percent are Asian-American and 16 percent are African-American.

From the ceiling of the student commons hang flags of countries from which students have come. There are more than 100. It’s a kind of kaleidoscope, and as I walked under it with Freeman Hrabowski, U.M.B.C.’s dynamic president, he stressed the school’s determination to “connect students to people different from themselves and lives different from their own.” The young men and women who ate, talked or studied at almost every one of the tables around us were a mix of colors, and I couldn’t map the room in terms of any obvious tribes or cliques.

Diversity, socioeconomic or otherwise, doesn’t factor much into U.S. News rankings, though a broadening of perspectives lies at the heart of the best education. U.M.B.C., with its acceptance rate of nearly 60 percent, places 159th among national universities.

One of the main factors in a school’s rank is how highly officials at peer institutions and secondary-school guidance counselors esteem it. But they may not know it well. They’re going by its reputation, established in no small part by previous U.S. News evaluations. A lofty rank perpetuates itself.

Another main factor is the percentage of a school’s students who graduate within six years. But this says as much about a school’s selectiveness — the proven achievement and discipline of the students it admits — as about its stewardship of them.


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abstract: 'Compressed sensing seeks to recover a sparse vector from a small number of linear and non-adaptive measurements. While most work so far focuses on Gaussian or Bernoulli random measurements we investigate the use of partial random circulant and Toeplitz matrices in connection with recovery by $\ell_1$-minization. In contrast to recent work in this direction we allow the use of an arbitrary subset of rows of a circulant and Toeplitz matrix. Our recovery result predicts that the necessary number of measurements to ensure sparse reconstruction by $\ell_1$-minimization with random partial circulant or Toeplitz matrices scales linearly in the sparsity up to a $\log$-factor in the ambient dimension. This represents a significant improvement over previous recovery results for such matrices. As a main tool for the proofs we use a new version of the non-commutative Khintchine inequality.'
author:
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title: Circulant and Toeplitz Matrices in Compressed Sensing
---

Introduction
============

Compressed sensing is a recent concept in signal processing where one seeks to reconstruct efficiently a sparse signal from a minimal number of linear and non-adaptive measurements [@do06-2]. So far various measurement matrices have been investigated, most of them random matrices. Among these are Bernoulli and Gaussian matrices [@badadewa06] (with independent $\pm 1$ or standard normal entries) as well as partial Fourier matrices [@cata06; @ra08-1; @ru06-1]. Recently, Bajwa et al. [@bahanorawr07] (see also [@ro08]) studied Toeplitz type and circulant matrices in the context of compressed sensing where the entries of the vector generating the Toeplitz or circulant matrix are chosen at random according to a suitable probability distribution. Compared to Bernoulli or Gaussian matrices random Toepliz and circulant matrices have the advantage that they require a reduced number of random numbers to be generated. More importantly, there are fast matrix-vector multiplication routines which can be exploited in recovery algorithms. Furthermore, they arise naturally in certain applications such as identifying a linear time-invariant system [@bahanora08].

Basis Pursuit ($\ell_1$-minimization) is one of the major approaches to efficiently recover a sparse vector. This technique is quite well understood by now. Modern optimization algorithms [@bova04] such as LARS [@efhajoti04] (sometimes called homotopy method) are reasonably fast.

Bajwa et al. [@bahanorawr07; @bahanora08] estimated the so-called restricted isometry constants of a random Toeplitz type or circulant matrix which then allows to provide recovery guarantees for $\ell_1$-minimization. However, their bound is very pessimistic compared to related estimates for Bernoulli / Gaussian or partial Fourier matrices. More precisely, the estimated number of measurements grows with the sparsity squared, while one would rather expect a linear scaling. Indeed, this is also suggested by numerical experiments. We close the theoretical gap by providing recovery guarantees for $\ell_1$-minimization in connection with circulant and Toeplitz type matrices where the necessary number of measurements scales linearly with the sparsity. However, we do not make use of the restricted isometry constants and a good estimate of the latter is therefore still open.

Sparse recovery with circulant and Toeplitz matrices
====================================================

For a vector $x\in {{\mathbb{R}}}^N$ we let ${\operatorname{supp}}x = \{j, x_j \neq 0 \}$ denote its support and $\|x\|_0 = |{\operatorname{supp}}x|$ the number of non-zero entries. It is called ${s}$-sparse if $\|x\|_0 \leq {s}$. We aim at recovering $x$ from $y= Ax \in {{\mathbb{R}}}^n$ where $A$ is a suitable $n \times N$ measurement matrix and $n < N$. A natural strategy is to consider $\ell_0$-minimization, $$\label{l0}
\min_x \|x\|_0 \quad \mbox{ subject to } Ax = y. $$ Unfortunately this combinatorial optimization problem is NP hard in general [@avdama97]. Therefore, we solve instead the convex problem $$\label{l1}
\min \|x\|_1 \quad \mbox{ subject to } Ax=y,$$ where the $\ell_p$-norm is defined as usual, $\|x\|_p= (\sum_{j=1}^N |x_j|^p)^{1/p}$. It is by now well understood that the solutions of both minimization problems often coincide and are equal to the original vector $x$, see e.g.  [@carota06; @dota06; @do06-2; @grni03; @ra07]. A by now popular result [@carota06; @ca08; @folaXX] states that indeed (stably) recovers all ${s}$-sparse $x$ from $y = Ax$ provided the restricted isometry constant $\delta_{2{s}} \leq \delta < \sqrt{2}-1$. The latter means that $$(1-\delta) \|x\|_2^2 \leq \|Ax\|_2^2 \leq (1+\delta) \|x\|_2^2$$ for all $2{s}$-sparse vectors $x$. It is known [@badadewa06] that random Gaussian or Bernoulli matrices, i.e. $n\times N$ matrices with independent and normal distributed or Bernoulli distributed entries, satisfy this condition with probability at least $1-\epsilon$ provided ${s}\leq C_1 n \log(N/{s})  + C_2 \log(\epsilon^{-1})$.

We consider the following types of measurement matrices. For $b = (b_0,b_1, \hdots,b_{N-1}) \in {{\mathbb{R}}}^N$ we let its associated circulant matrix $S = S^b \in {{\mathbb{R}}}^{N\times N}$ with entries $S_{i,j} = b_{j-i \mod N}$, where $i,j =1,\hdots,N$. Similarly, for a vector $c = (c_{-N+1},c_{-N+2},\hdots,c_{N-1})$ its associated Toeplitz matrix $T = T ^c \in {{\mathbb{R}}}^{N \times N}$ has entries $T_{i,j} = c_{j-i}$, where $i,j =1,\hdots,N$. Now we choose an arbitrary subset $\Omega \subset \{1,\hdots,N\}$ of cardinality $n<N$ and let the partial circulant matrix $S_\Omega = S_\Omega^b \in {{\mathbb{R}}}^{n \times N}$ be the submatrix of $S$ consisting of the rows indexed by $\Omega$. The partial Toeplitz matrix $T_\Omega=T_\Omega^c \in {{\mathbb{R}}}^{n \times N}$ is defined similarly. In this paper the vectors $b$ and $c$ will always be random vectors with independent Bernoulli $\pm 1$ entries.

Of particular interest is the case $N = n K$ for some $K \in {{\mathbb{N}}}$ and $\Omega = \{ K, 2K, \hdots, nK\}$. Then the application of $S_\Omega^b$ and $T_\Omega^c$ corresponds to (periodic or non-periodic) convolution with the sequence $b$ (or $c$, respectively) followed by a downsampling by a factor of $K$. This setting was studied numerically in [@babadutrwa06] by Tropp et al. (using orthogonal matching pursuit instead of $\ell_1$-minimization). Also of interest is the case $\Omega = \{1,2,\hdots,n\}$ which was investigated in [@bahanorawr07; @bahanora08] by Bajwa et al., who showed that the restricted isometry constant of $T_\Omega^c$ satisfies $\delta_{{s}} \leq \delta$ with high probability (w.h.p.) provided $n \geq C_\delta {s}^2 \log(N/{s})$. As a byproduct of the proof of our main result we give an alternative proof that $\delta_{s}\leq \delta$ holds w.h.p. under the condition $n \geq C \delta^{-2} {s}^2 \log^2(N)$. However, we strongly believe that this bound is not optimal due to the quite pessimistic quadratic scaling in ${s}$. Our main result shows that one can achieve recovery w.h.p. by $\ell_1$-minimization, if $n \geq C {s}\log^2(N)$.

In the following recovery theorem we use a random partial circulant or Toeplitz matrix $A^b_\Omega$ or $T^c_\Omega$ in the sense that the entries of the vector $b$ or $c$ are independent Bernoulli $\pm 1$ random variables. Furthermore, the signs of the non-zero entries of the ${s}$-sparse vector $x$ are chosen at random according to a Bernoulli distribution as well. In contrast to previous work [@bahanorawr07; @babadutrwa06] $\Omega$ is allowed to be an arbitrary subset of $\{1,\hdots,N\}$ of cardinality $n$.

\[thm:main\] Let $\Omega \subset \{1,2,\hdots,N\}$ be an arbitrary (deterministic) set of cardinality $n$. Let $x\in{{\mathbb{R}}}^N$ be ${s}$-sparse such that the signs of its non-zero entries are Bernoulli $\pm 1$ random variables. Choose $b \in {{\mathbb{R}}}^N$ to be a random vector whose entries are $\pm 1$ Bernoulli variables. Let $y = S^b_\Omega x \in {{\mathbb{R}}}^n$. There exists a constant $C > 0$ such that $$n \geq C {s}\log^3(N/\epsilon)$$ implies that with probability at least $1-\epsilon$ the solution of the $\ell_1$-minimization problem coincides with $x$.

The same statement holds with $T^c_\Omega$ in place of $S^b_\Omega$ where $c \in {{\mathbb{R}}}^{2N-1}$ is a random vector with Bernoulli $\pm 1$ entries.

Ignoring the $\log$-factor the necessary number of samples ensuring recovery by $\ell_1$-minimization scales linearly with the sparsity ${s}$. The power $3$ at the $\log$-term can very likely be improved to $1$, and moreover, it seems also possible to remove the randomness assumption on the non-zero coefficients of $x$. We postpone such improvements as well as an investigation of the restricted isometry constants to possible future contributions. The remainder of the paper is concerned with the proof of Theorem \[thm:main\].

Proof of Theorem \[thm:main\]
=============================

An essential ingredient of the proof is the following recovery theorem for $\ell_1$-minimization due to Fuchs [@fu04] and Tropp [@tr05-1]. For a matrix $A$ we denote by $a_\rho$ its columns and by $A_\Lambda$ the submatrix consisting only of the columns index by $\Lambda$.

\[thm:Tropp\] Suppose that $y = A x$ for some $x$ with ${\operatorname{supp}}x = \Lambda$. If $$\label{recovery:cond}
|\langle A_\Lambda^\dagger {{a}}_\rho,  {\operatorname{sgn}}(x_\Lambda) \rangle | < 1 \quad \mbox{ for all } \rho \notin \Lambda\,,$$ then $x$ is the unique solution of the Basis Pursuit problem (\[l1\]). Here, $A_\Lambda^\dagger$ denotes the Moore-Penrose pseudo-inverse of $A_\Lambda$.

A crucial step in applying this theorem is to show that the $\ell_2$-norm of ${{A}}_\Lambda^\dagger {{a}}_\rho$ in is small. To this end one expands $$\label{PseudoInvEst}
\|{{A}}_\Lambda^\dagger {{a}}_\rho\|_2 \,=\, \|({{A}}_\Lambda^* {{A}}_\Lambda)^{-1} {{A}}_\Lambda^* {{a}}_\rho\|_2 \,=\, \|({{A}}_\Lambda^* {{A}}_\Lambda)^{-1}\|_{2\to 2} \|{{A}}_\Lambda^* {{a}}_\rho\|_2,$$ where $\|\cdot\|_{2\to2}$ denotes the operator norm on $\ell_2$. The second term can be estimated in terms of the coherence of ${{A}}$, which is defined to be the largest absolute inner product of different columns of ${{A}}$, $\mu = \max_{\rho \neq \lambda} |\langle {{a}}_\rho, {{a}}_\lambda \rangle|$. Indeed, $$\|{{A}}_\Lambda^* {{a}}_\lambda\|_2 =
\left( \sum_{\lambda \in \Lambda} |\langle {{a}}_{\lambda}, {{a}}_\rho \rangle|^2 \right)^{1/2}
\leq \sqrt{|\Lambda|} \mu.$$ The coherence of a random Toeplitz or circulant matrix can be bounded as follows.

\[prop:coherence\] Let $\mu$ be the coherence of the random partial circulant matrix $\frac{1}{\sqrt{n}} S_\Omega^b \in {{\mathbb{R}}}^{n\times N}$ or Toeplitz matrix $\frac{1}{\sqrt{n}} T_\Omega^c \in {{\mathbb{R}}}^{n\times N}$ where $b$ and $c$ are Rademacher series and $\Omega$ has cardinality $n$. Then with probability at least $1-\epsilon$ the coherence satisfies $$\mu \leq 4 \frac{\log(2N^2/\epsilon)}{\sqrt{n}}.$$

The proof is contained in Section \[sec:proof:coherence\]. This proposition easily implies the following (probably non-optimal) estimate of the restricted isometry constants of $S^b_\Omega$ or $T^c_\Omega$ contained also in [@bahanora08] with a different proof.

\[cor\_RIP\] Let $\frac{1}{\sqrt{n}} S_\Omega^b, \frac{1}{\sqrt{n}} T_\Omega^c \in {{\mathbb{R}}}^{n\times N}$ be the randomly generated normalized partial circulant and Toeplitz matrix generated from Rademacher series and $\delta_{s}$ be their restricted isometry constant. Assume that $$n \geq 16 \delta^{-2} {s}^2 \log^2(2N^2/\epsilon).$$ Then with probability at least $1-\epsilon$ it holds $\delta_{s}\leq \delta$.

Combine the bound $\delta_{s}\leq ({s}-1) \mu$ (which easily follows from Gershgorin’s disk theorem) with the estimate above on the coherence of $A = \frac{1}{\sqrt{n}} S^b_\Omega$ or $A = \frac{1}{\sqrt{n}} T^c_\Omega$.

As suggested by we also need an estimate of the operator norm of the inverse of $A_\Lambda^* A_\Lambda$. To this end we bound the smallest and largest eigenvalue of this matrix.

\[thm:condition\] Let $\Omega,\Lambda \subset \{1,\hdots,N\}$ with $|\Omega| =n$ and $|\Lambda| = {s}$. Let $b \in {{\mathbb{R}}}^N$ and $c \in {{\mathbb{R}}}^{2N-1}$ be Rademacher series. Denote either $A = \frac{1}{\sqrt{n}} S^b_\Omega$ or $A = \frac{1}{\sqrt{n}} T^c_\Omega$. Assume $$\label{cond:conditioning}
n \geq \tilde{C} \delta^{-2} {s}\log^2(4{s}/\epsilon),$$ where $\tilde{C} = 4\pi^2 \approx 39.48$. Then with probability at least $1- \epsilon$ the minimal and maximal eigenvalues $\lambda_{\min}$ and $\lambda_{\max}$ of $A_\Lambda^* A_\Lambda$ satisfy $$1-\delta \leq \lambda_{\min} \leq \lambda_{\max} \leq 1+\delta.$$

Note that the above theorem holds for a fixed subset $\Lambda$ and random coefficients $b$ or $c$. It does not imply that for given $b$ or $c$ the estimate holds uniformly for all subsets $\Lambda$, which would be equivalent to having an estimate for the restricted isometry constants of $\frac{1}{\sqrt{n}} S^b_\Omega$ or $\frac{1}{\sqrt{n}} T^c_\Omega$. (Note that taking a union bound over all subsets $\Lambda$ would yield an estimate essentially worse than Corollary \[cor\_RIP\].)

Now we are ready to complete the proof of Theorem \[thm:main\] on the basis of Proposition \[prop:coherence\] and Theorem \[thm:condition\]. We proceed similarly as in [@tr06-2 Theorem 14]. Hoeffding’s inequality states that $$\label{Bernstein}
{{\mathbb{P}}}\big(|\sum_{j} \epsilon_j a_j|\geq u \|a\|_2\big) \leq 2 e^{- u^2/2}.$$ By our assumption on the random phases $\epsilon_\lambda = {\operatorname{sgn}}(x_\lambda)$, the scalar product on the left hand side of (\[recovery:cond\]) is precisely of the above form with $a = A_\Lambda^\dagger a_\rho = (A_\Lambda^* A_\Lambda)^{-1} A_\Lambda^* a_\rho$. Theorem \[thm:condition\] implies that the smallest eigenvalue of $A_\Lambda^* A_\Lambda$ is bounded from below by $1-\delta$ with probability at least $1-\epsilon$ provided condition holds; hence, $\|(A_\Lambda^* A_\Lambda)^{-1}\|_{2\to 2} \leq \frac{1}{1-\delta}$. Plugging this into yields $$\label{equation:CombineEstimates1}
\|{{A}}_\Lambda^\dagger {{a}}_\rho\|_2 \leq \frac{1}{1-\delta} \sqrt{{s}} \mu.$$ Following Theorem \[thm:Tropp\] the probability that recovery fails can be estimated by $$\begin{aligned}
&{{\mathbb{P}}}\big(|\langle {{A}}_\Lambda^\dagger {{a}}_\rho, R_\Lambda {\operatorname{sgn}}(x) \rangle| \geq 1
\mbox{ for some } \rho \notin \Lambda\big)\notag\\[.2cm]
&\leq {{\mathbb{P}}}\big(\,|\langle {{A}}_\Lambda^\dagger {{a}}_\rho, R_\Lambda {\operatorname{sgn}}(x) \rangle| \geq 1
\mbox{ for some } \rho \notin \Lambda\ \big|
 \mu \leq \frac{\alpha}{\sqrt{n}} \notag\\
& \phantom{\leq {{\mathbb{P}}}\big( }
\mbox{\&}\, \lambda_{\min} \geq 1-\delta\,\big) + {{\mathbb{P}}}\big(\mu > \frac{\alpha}{\sqrt{n}}\big) + {{\mathbb{P}}}\big(\lambda_{\min} < 1-\delta\big) \notag\\[.2cm]
&\leq \sum_{\rho \notin \Lambda} {{\mathbb{P}}}\big(\,|\langle {{A}}_\Lambda^\dagger {{a}}_\rho, R_\Lambda {\operatorname{sgn}}(x) \rangle| \geq 1\ \big| \ \mu \leq \frac{\alpha}{\sqrt{n}} \, \mbox{\&}\, \lambda_{\min} \geq 1-\delta\,\big) \notag\\[.2cm]&\qquad+
{{\mathbb{P}}}\big(\mu > \frac{\alpha}{\sqrt{n}}\big) + {{\mathbb{P}}}\big(\lambda_{\min} < 1-\delta\big).\notag\end{aligned}$$ Under the assumption $\mu \leq \frac{\alpha}{\sqrt{n}}$ equation implies that for $u= \frac {(1-\delta)\sqrt n}{\alpha \sqrt{{s}}}$ we have $u\| A_\Lambda^\dagger a_\rho\|_2\leq 1$, so (\[Bernstein\]) gives $$\begin{aligned}
\label{equation:CombineEstimates2}
&{{\mathbb{P}}}\big(\,|\langle {{A}}_\Lambda^\dagger {{a}}_\rho, R_\Lambda {\operatorname{sgn}}(x) \rangle| \geq 1\ \big|\ \mu \leq \frac{\alpha}{\sqrt{n}}\,  \mbox{\&} \, \lambda_{\min} \geq 1-\delta\, \big)\notag\\
&\leq \ 2 \exp\left(-\frac{(1-\delta)^2}{2\alpha^2} \frac{n}{{s}}\right).\end{aligned}$$ Setting $\alpha = 4 \log(2N^2/\epsilon)
$ Theorem \[prop:coherence\] yields $${{\mathbb{P}}}(\mu \geq \alpha/\sqrt{n}) \leq \epsilon.$$ Now we choose $\delta = 1/2$. Under condition (\[cond:conditioning\]), which reads $$\label{cond2}
n \geq 4\tilde{C} {s}\log^2({s}/\epsilon),$$ we have ${{\mathbb{P}}}(\lambda_{\min} \geq 1-\delta) \leq \epsilon$. Hence, under the above conditions we obtain $$\begin{aligned}
&{{\mathbb{P}}}\big(|\langle {{A}}_\Lambda^\dagger {{a}}_\rho, R_\Lambda {\operatorname{sgn}}(x) \rangle| \geq 1
\mbox{ for some } \rho \notin \Lambda\big) \notag\\
&\leq 2 N \exp\left(- \frac{1}{8\log^2(2N^2/\epsilon)}\frac{n}{{s}}\right) + 2\epsilon.\label{P_estimate1}\end{aligned}$$ The first term is less than $\epsilon$ provided $n \geq 8 {s}\log^2(2N^2/\epsilon)\log(2 N/\epsilon)$, or $$\label{cond1}
n \geq C_1 {s}\log^3(N/\epsilon)$$ for a suitable constant $C_1$. Conditions (\[cond2\]) and (\[cond1\]) are both satisfied if $$n \geq C {s}\log^3(N/\epsilon)$$ for a suitable constant $C$, in which case the probability that recovery by $\ell_1$-minimization is less than $3 \epsilon$. This completes the proof.

Non-commutative Khintchine inequalities
=======================================

Both the proof of Proposition \[prop:coherence\] as well as the proof of Theorem \[thm:condition\] are based on versions of the Khintchine inequality. Let us first state the non-commutative Khintchine inequality due to Lust-Piquard [@lu86-1] and Buchholz [@bu01], see also [@tr06-2]. To this end we introduce Schatten class norms on matrices. Denoting by $\sigma(A)$ the vector of singular values of a matrix $A$, the $S_p$-norm is defined as $$\|A\|_{S_p} := \|\sigma(A)\|_p,$$ where $\|\cdot\|_p$ is the usual $\ell_p$-norm, $1\leq p\leq \infty$.

\[Gauss\_Khintchine\] Let $(A_k)$ be a finite sequence of matrices of the same dimension and let $(g_k)$ be a sequence of independent standard Gaussian random variables. Then for $m \in {{\mathbb{N}}}$, $$\begin{aligned}
&\left[ {{\mathbb{E}}}\left\|\sum_k g_k A_k\right\|_{S_{2m}}^{2m} \right]^{1/2m} \notag\\
&\leq B_m \max \left\{ \left\|\left(\sum_k A_k A_k^*\right)^{1/2}\right\|_{S_{2m}}, \left\| \left(\sum_k A_k^* A_k \right)^{1/2} \right\|_{S_{2m}} \right\},\notag\end{aligned}$$ with optimal constant $$B_m = \left( \frac{(2m)!}{2^m m!}\right)^{\frac{1}{2m}}$$

Using the contraction principle for Bernoulli random variables, see [@leta91 eq. (4.8)], we obtain the non-commutative Khintchine inequality for Bernoulli random variables [@lu86-1].

\[cor\_Kh\] Let $(A_k)$ be a finite sequence of matrices of the same dimension and let $(\epsilon_k)$ be a sequence of independent Bernoulli $\pm 1$ random variables. Then for $m \in {{\mathbb{N}}}$, $$\begin{aligned}
&\left[ {{\mathbb{E}}}\left\|\sum_k \epsilon_k A_k\right\|_{S_{2m}}^{2m} \right]^{1/2m} \notag\\
&\leq C_m \max \left\{ \left\|\left(\sum_k A_k A_k^*\right)^{1/2}\right\|_{S_{2m}}, \left\| \left(\sum_k A_k^* A_k \right)^{1/2} \right\|_{S_{2m}} \right\},\label{Bernoulli_Khintch}\end{aligned}$$ with constant $$C_m = \sqrt{\frac{\pi}{2}} \left( \frac{(2m)!}{2^m m!}\right)^{\frac{1}{2m}}.$$

In the scalar case the factor $\sqrt{\pi/2}$ can be removed. However, it is not clear yet whether this is true also in the non-commutative situation.

The following theorem extends the non-commutative Khintchine inequality to a second order chaos variable. Its proof uses decoupling and Corollary \[cor\_Kh\].

\[Bernoulli\_Khintchine\] Let $A_{j,k} \in {{\mathbb{C}}}^{r \times t}$, $j,k=1,\hdots,N$, be matrices with $A_{j,j} = 0$, $j=1,\hdots,N$. Let $\epsilon_k$, $k=1, \hdots, N$ be independent Bernoulli random variables. Then for $m \in {{\mathbb{N}}}$ it holds $$\begin{aligned}
&\left[ {{\mathbb{E}}}\left\| \sum_{j,k=1}^N \epsilon_j \epsilon_k A_{j,k} \right\|_{S_{2m}}^{2m}\right]^{1/2m} \notag\\
&\leq D_m \max \left\{ \left\|\left(\sum_{j,k=1}^N A_{j,k} A_{j,k}^*\right)^{1/2}\right\|_{S_{2m}},\right.\notag\\
& \phantom{\leq D_m \max \{\{ }
\left. \left\| \left( \sum_{j,k=1}^N A_{j,k}^* A_{j,k}\right)^{1/2}\right\|_{S_{2m}}, \|F\|_{S_{2m}}\right\},   \notag\end{aligned}$$ where $F$ is the block matrix $F = (A_{j,k})_{j,k=1}^N$ and the constant $$D_m \,=\, 2^{1/2m} 2\pi C_m^2 = 2^{1/2m} 2\pi \left(\frac{(2m)!}{2^m m!}\right)^{1/m}.$$

At present it is not clear whether the term $\|F\|_{S_{2m}}$ can be omitted above. At least, there is no a priori inequality between any of the terms in the maximum. The proof of the theorem is based on the following decoupling lemma, see [@botz87 Proposition 1.9] or [@gide99 Theorem 3.1.1].

\[lem\_decouple\] Let $\xi_j$, $j=1,\hdots,N$, be a sequence of independent random variables with ${{\mathbb{E}}}\xi_j = 0$ for all $j=1,\hdots,N$. Let $A_{j,k}$, $j,k=1,\hdots,N$, be a double sequence of elements in a Banach space with norm $\|\cdot\|$, where $A_{j,j} = 0$ for all $j=1,\hdots,N$. Then for $1\leq p < \infty$ $${{\mathbb{E}}}\left\| \sum_{j,k=1}^N \xi_j \xi_k A_{j,k} \right\|^p
\leq 4^p {{\mathbb{E}}}\left\| \sum_{j,k=1}^N \xi_j \xi'_k A_{j,k} \right\|^p,$$ where $\xi'$ denotes an independent copy of the sequence $\xi = (\xi_j)$.

[Proof of Theorem \[Bernoulli\_Khintchine\]]{}. We apply Lemma \[lem\_decouple\] followed by the non-commutative Khintchine inequality , $$\begin{aligned}
E& := {{\mathbb{E}}}\left\| \sum_{j,k=1}^N \epsilon_j \epsilon_k A_{j,k} \right\|_{S_{2m}}^{2m}\notag\\
&\leq 4^{2m} {{\mathbb{E}}}_{\epsilon} {{\mathbb{E}}}_{\epsilon'} \left\| \sum_{j,k=1}^N \epsilon_j \epsilon_k' A_{j,k} \right\|_{S_{2m}}^{2m}\notag\\
&\leq 4^{2m} C_m^{2m} {{\mathbb{E}}}_{\epsilon} \max\left\{ \left\|\left(\sum_{k=1}^N B_k(\epsilon)^* B_k(\epsilon)\right)^{1/2} \right\|_{S_{2m}}^{2m}, \right.\notag\\
&\phantom{\leq 4^{2m} C_m^{2m} } \left.
\left\| \left(\sum_{k=1}^N B_k(\epsilon) B_k(\epsilon)^*\right)^{1/2}
\right\|_{S_{2m}}^{2m} \right\},\label{MaxEst}\end{aligned}$$ where $B_k(\epsilon) \,:=\, \sum_{j=1}^N \epsilon_j A_{j,k}$. We define $$\begin{aligned}
\widehat{A}_{j,k} = \left(0|\hdots|0|A_{j,k}|0|\hdots|0\right) \in {{\mathbb{C}}}^{r \times tN}\notag\end{aligned}$$ where the non-zero block $A_{j,k}$ is the $k$-th one, and similarly $$\begin{aligned}
\widetilde{A}_{j,k} = \left(0|\hdots|0|A_{j,k}^*|0|\hdots|0\right)^* \in {{\mathbb{C}}}^{rN \times t}.\notag\end{aligned}$$ Then clearly $$\begin{aligned}
\widehat{A}_{j,k} \widehat{A}^*_{j',k'} = \left\{\begin{array}{cc} 0 &  \mbox{ if } k \neq k',\\
A_{j,k} A_{j',k}^* & \mbox{ if } k = k'.
\end{array} \right.,\label{wideAA}\\
\widetilde{A}_{j,k}^* \widetilde{A}_{j',k'} = \left\{\begin{array}{cc} 0 &  \mbox{ if } k \neq k',\\
A_{j,k}^* A_{j',k} & \mbox{ if } k = k'.
\end{array} \right.\notag\end{aligned}$$ The Schatten class norm satisfies $\| A \|_{S_{2m}} = \|(A A^*)^{1/2}\|_{S_{2m}}$. This allows us to verify that $$\begin{aligned}
&\left\|\sum_{j=1}^N \epsilon_j \sum_{k=1}^N \widehat{A}_{j,k} \right\|_{S_{2m}}
\,=\, \left\|\left(\sum_{j,j'} \epsilon_j \epsilon_{j'} \sum_{k,k'} \widehat{A}_{j,k} \widehat{A}_{j',k'}^*\right)^{1/2}\right\|_{S_{2m}}\notag\\
&=\, \left\|\left(\sum_{j,j'} \epsilon_j \epsilon_{j'} \sum_k A_{j,k} A_{j',k}^* \right)^{1/2} \right\|_{S_{2m}}\notag\\
&=\, \left\|\left(\sum_k B_k(\epsilon) B_k(\epsilon)^*\right)^{1/2}\right\|_{S_{2m}}.\notag\end{aligned}$$ Similarly, we also verify that $$\left\|\left(\sum_k B_k(\epsilon)^* B_k(\epsilon)\right)^{1/2}\right\|_{S_{2m}} = \left\|\sum_{j=1}^N \epsilon_j \sum_{k=1}^N \widetilde{A}_{j,k}\right\|_{S_{2m}}.$$ Plugging the above expressions into (\[MaxEst\]) we can further estimate $$\begin{aligned}
E &\leq 4^{2m} C_{2m}^{2m} \left({{\mathbb{E}}}\left\| \sum_{j=1}^N \epsilon_j \sum_{k=1}^N \widehat{A}_{j,k}\right\|_{S_{2m}}^{2m}\right.\notag\\
& \phantom{\leq 4^{2m} C_{2m}^{2m} ((}
\left. + {{\mathbb{E}}}\left\| \sum_{j=1}^N \epsilon_j \sum_{k=1}^N \widetilde{A}_{j,k}\right\|_{S_{2m}}^{2m}\right).\notag\end{aligned}$$ Using Khintchine’s inequality once more we obtain $$\begin{aligned}
E_1\,&:=
{{\mathbb{E}}}\left\| \sum_{j=1}^N \epsilon_j \sum_{k=1}^N \widehat{A}_{j,k}\right\|_{S_{2m}}^{2m}\notag\\
&\leq C_m^{2m}
\max\left\{ \left\| \left( \sum_j \widetilde{B}_j \widetilde{B}_j^*\right)^{1/2} \right\|_{S_{2m}}^{2m},\right.\notag\\
&\phantom{\leq C_m^{2m} \max \{\{} \left.
\left\| \left( \sum_j \widetilde{B}_j^* \widetilde{B}_j\right)^{1/2} \right\|_{S_{2m}}^{2m} \right\}, \notag\end{aligned}$$ where $\widetilde{B}_j \,=\, \sum_{k=1}^N \widehat{A}_{j,k}$. Using (\[wideAA\]) we see that $$\sum_j \widetilde{B}_j \widetilde{B}_j^* = \sum_{k,j} A_{j,k} A_{j,k}^*.$$ Furthermore, with the block matrix $$\begin{aligned}
F = \left(\begin{array}{cc}\widetilde{B}_1\\\widetilde{B}_2\\ \vdots \\
\widetilde{B}_N\end{array}\right)
\,=\, \left(\begin{array}{cccc}
A_{1,1} & A_{1,2} & \hdots & A_{1,N}\\
A_{2,1} & A_{2,2} & \hdots & A_{2,N}\\
\vdots & \vdots & \vdots & \vdots \\
A_{N,1} & A_{N,2} & \hdots & A_{N,N} \\
\end{array}\right) \notag\end{aligned}$$ we have $$\|(\sum_k \widetilde{B}_k^* \widetilde{B}_k)^{1/2}\|_{S_{2m}}^{2m}
\,=\, \|(F^* F)^{1/2} \|_{S_{2m}}^{2m} \,=\, \|F\|_{S_{2m}}^{2m}.$$ Hence, $$E_1 \leq C_m^{2m} \max\left\{ \|(\sum_{j,k=1}^N A_{j,k} A_{j,k}^*)^{1/2}\|_{S_{2m}}^{2m}, \|F\|_{S_{2m}}^{2m}\right\}.$$ As $\widetilde{A}_{j,k}$ differs from $\widehat{A}_{j,k}$ only by interchanging $A_{j,k}$ with $A_{j,k}^*$ we obtain similarly $$\begin{aligned}
&E_2 := {{\mathbb{E}}}\left\| \sum_{j=1}^N \epsilon_j \sum_{k=1}^N \widetilde{A}_{j,k} \right\|_{S_{2m}}^{2m}\notag\\
&\leq \max\left\{\left\|\left(\sum_{j,k=1}^N A_{j,k}^* A_{j,k} \right)^{1/2}\right\|_{S_{2m}}^{2m},\|F\|_{S_{2m}}^{2m} \right\}. \notag\end{aligned}$$ Finally, we obtain $$\begin{aligned}
& E \leq 4^{2m} C_m^{2m} (E_1 + E_2) \notag\\
& \leq 2 \cdot 4^{2m} C_m^{4m} \max\left\{\left\|\left(\sum_{j,k=1}^N A_{j,k}^* A_{j,k} \right)^{1/2}\right\|_{S_{2m}}^{2m},\right.\notag\\
& \phantom{\leq 2 \cdot 4^{2m} C_m^{4m} }
\left.
\left\|\left(\sum_{j,k=1}^N A_{j,k} A_{j,k}^* \right)^{1/2}\right\|_{S_{2m}}^{2m},\|F\|_{S_{2m}}^{2m}\right\}.\notag\end{aligned}$$ This concludes the proof.

Repeating the above proof for the scalar case (which removes the factor $\pi/2$ in the constant) and applying interpolation (see and below) yields the following (compare also [@mcta86 Proposition 2.2]).

\[Scalar\_Khintchine\] Let $a_{j,k} \in {{\mathbb{C}}}$, $j,k=1,\hdots,N$ be numbers with $a_{j,j} = 0$, $j=1,\hdots,N$. Let $\epsilon_k$, $k=1, \hdots, N$ be independent Bernoulli $\pm 1$ random variables. Then for $2 \leq p < \infty$ it holds $$\left[ {{\mathbb{E}}}\left| \sum_{j,k=1}^N \epsilon_j \epsilon_k a_{j,k} \right|^{p}\right]^{1/p}
\leq d_p \left(\sum_{j,k=1}^N |a_{j,k}|^2\right)^{1/2},$$ where the constant $$d_p \,=\, 4^{1/p} (4/e) p.$$

Proof of the coherence estimate {#sec:proof:coherence}
===============================

Now we are equipped to provide the proof of Proposition \[prop:coherence\]. An inner product of two columns $s_i, s_\ell$ of the normalized matrix $\frac{1}{\sqrt{n}} S^b_\Omega$ has the form $$\langle s_i, s_\ell \rangle \,=\, \frac{1}{n} \sum_{r\in \Omega} b_{i - r \mod N} b_{\ell - r \mod N}
\,=\, \frac{1}{n} \sum_{j,k=1}^N b_{j} b_k a_{j,k}^{i,\ell},$$ where $a_{j,k}^{i,\ell} = 1$ if $(j,k) = (i - r \mod N, \ell-r \mod N)$ for some $r \in \Omega$ and $a_{j,k}^{i,\ell} = 0$ otherwise. Similarly, the inner product of the columns $t_i$ of the normalized matrix $\frac{1}{\sqrt{n}} T^c_\Omega$ can be written as $
\langle t_i, t_\ell \rangle = n^{-1} \sum_{j,k=-N+1}^{N-1} c_{j} c_k \tilde{a}_{j,k}^{i,\ell}
$ with $\tilde{a}_{j,k}^{i,\ell} = 1$ if $(j,k) = (i-r,\ell-r) \in \{1,\hdots,N\}^2$ for some $r \in \Omega$ and $0$ otherwise. Observe that $\sum_{j,k} |a_{j,k}|^2 = \sum_{j,k} |\tilde{a}_{j,k}|^2 = |\Omega| =n$. Now let $b\in {{\mathbb{R}}}^N$ and $c \in {{\mathbb{R}}}^{2N}$ be Rademacher series. Then Corollary \[Scalar\_Khintchine\] yields $$\begin{aligned}
&n\left({{\mathbb{E}}}|\langle s_i, s_j\rangle|^p\right)^{1/p}=  \left({{\mathbb{E}}}|\sum_{j,k} b_j b_k a_{j,k}^{i,\ell}|^p\right)^{1/p}\notag\\
&\leq 4^{1/p} (4/e) p  \left(\sum_{j,k} |a_{j,k}|^2 \right)^{1/2}= 4^{1/p} (4/e) p \sqrt{n}\notag\end{aligned}$$ for $p\geq 2$, and the same estimate holds for ${{\mathbb{E}}}|\langle t_i, t_j\rangle|^p$. In order to complete the proof we use the following simple and well-known probability estimate, see e.g. [@leta91; @tr06-2].

\[lem\_moment1\] Suppose $Z$ is a positive random variable satisfying $({{\mathbb{E}}}Z^p)^{1/p} \leq \alpha \beta^{1/p} p^{1/\gamma}$ for all $p_0 \leq p < \infty$ and some $\alpha,\beta,\gamma > 0$. Then for arbitrary $\kappa > 0$, $${{\mathbb{P}}}(Z \geq e^\kappa \alpha u) \leq \beta e^{-\kappa u^\gamma}$$ for all $u \geq p_0$.

By Markov’s inequality we obtain $${{\mathbb{P}}}(Z \geq e^\kappa \alpha u)
\leq \frac{{{\mathbb{E}}}Z^p}{(e^\kappa \alpha u)^p}
\leq  \beta \left(\frac{\alpha p^{1/\gamma}}{e^\kappa \alpha u}\right)^p .$$ Choosing $p=u^\gamma$ yields the statement.

Lemma \[lem\_moment1\] with the optimal choice $\kappa = 1$ yields $${{\mathbb{P}}}(n |\langle s_i,s_\ell \rangle| \geq  4 \sqrt{n} u)
\leq 4 e^{- u}$$ for $u\geq 2$. Taking the union bound over all possible pairs of different columns $s_i, s_\ell$ we obtain $$\begin{aligned}
{{\mathbb{P}}}(\mu \geq  4 n^{-1/2} u) \leq 2 N^2 e^{-u}.\notag\end{aligned}$$ Set the right hand side to $\epsilon$. Then the resulting $u = \log(2N^2/\epsilon) \geq 2$ since we may assume without loss of generality that $N\geq 2$. We obtain $${{\mathbb{P}}}\left(\mu \geq 4 \frac{\log(2N^2/\epsilon)}{\sqrt{n}}\right) \leq \epsilon.$$ The same holds for the coherence of $\frac{1}{\sqrt{n}} T_\Omega^c$.

Proof of Theorem \[thm:condition\]
==================================

We introduce the elementary shift operators on ${{\mathbb{R}}}^N$, $(S_j x)_\ell = x_{\ell - j \mod N}$, $j=1,\hdots,N$, and $$(T_j x)_\ell = \left\{\begin{array}{ll} x_{\ell-j} & \mbox{if } 1 \leq \ell-j \leq N, \\
0 & \mbox{otherwise},
\end{array} \right.$$ for $j = -N+1,\hdots,N-1$, $\ell=1,\hdots,N$. Further, denote by $R_\Omega: {{\mathbb{R}}}^N \to {{\mathbb{R}}}^\Omega$ the operator that restricts a vector to the indices in $\Omega$. Then we can write $$\begin{aligned}
S_\Omega^b = R_\Omega \sum_{j=1}^N \epsilon_j S_j \quad \mbox{and} \quad
T_\Omega^c  = R_\Omega \sum_{j=-N+1}^{N-1} \epsilon_j T_j, \notag\end{aligned}$$ where $(\epsilon_j)$ is a Rademacher sequence. Denote by $A$ either $\frac{1}{\sqrt{n}} S_\Omega^b$ or $\frac{1}{\sqrt{n}} T_\Omega^c$. We need to prove a bound on the operator norm of $X_\Lambda := A_\Lambda^* A_\Lambda - I_\Lambda$ where $I_\Lambda$ denotes the identity on ${{\mathbb{R}}}^\Lambda$. We introduce $R_\Lambda^*: {{\mathbb{R}}}^\Lambda \to {{\mathbb{R}}}^N$ to be the extension operator that fills up a vector in ${{\mathbb{R}}}^\Lambda$ with zeros outside $\Lambda$. Further, we denote by $D_j$ either $S_j$ or $T_j$. Observe that $$\begin{aligned}
&A_\Lambda^* A_\Lambda \, =\, \frac{1}{n} \sum_{j}\epsilon_j R_\Lambda D_j^* R_\Omega^* \sum_{k} \epsilon_{k} R_\Omega D_{k} R_\Lambda^*\notag\\
&=\, \frac{1}{n} \sum_{\substack{j,k\\j\neq k}} \epsilon_j \epsilon_{k} R_\Lambda D_j^* P_\Omega D_{k} R_\Lambda^* + \frac{1}{n} R_\Lambda \left(\sum_{j} D_j^* P_\Omega D_j\right) R_\Lambda^*, \notag\end{aligned}$$ where $P_\Omega = R_\Omega^* R_\Omega$ denotes the projection operator which cancels all components of a vector outside $\Omega$. Here and in the following the sums range either over $\{1,\hdots,N\}$ or over $\{-N+1,\hdots,N-1\}$ depending on whether we consider circulant or Toeplitz matrices. It is straightforward to check that $$\label{sum_Khintch}
\sum_{j} D_j^* P_\Omega D_j = n I_N,$$ where $I_N$ is the identity on ${{\mathbb{R}}}^N$. Since $R_\Lambda R_\Lambda^* = I_\Lambda$ we obtain $$X_\Lambda = \frac{1}{n} \sum_{j\neq k} \epsilon_j \epsilon_i R_\Lambda D_j^* P_\Omega D_k R_\Lambda^* = \frac{1}{n} \sum_{j \neq k} \epsilon_j \epsilon_k A_{j,k}$$ with $A_{j,k} = R_\Lambda D_j^* P_\Omega D_k R_\Lambda^*$. Our goal is to apply Corollary \[Bernoulli\_Khintchine\]. To this end we first observe that by $$\begin{aligned}
\sum_j A_{j,k}^* A_{j,\ell} \, & = \, R_\Lambda D_k^* P_\Omega \left( \sum_j D_j P_\Lambda D_j^*\right) P_\Omega D_\ell R_\Lambda^*\notag\\
&=\, {s}R_\Lambda D_k^* P_\Omega  D_\ell R_\Lambda^*. \notag\end{aligned}$$ Using once more this yields $$\sum_{j,k} A_{j,k}^* A_{j,k} = {s}R_\Lambda \left(\sum_{k} D_k^* P_\Omega D_k\right) R_\Lambda^* = {s}n R_\Lambda R_\Lambda^* = {s}n I_\Lambda.$$ Since the entries of all matrices $A_{j,k}$ are non-negative we get $$\begin{aligned}
& \|(\sum_{j\neq k} A_{j,k}^* A_{j,k})^{1/2}\|_{S_{2m}}^{2m}  = {\operatorname{Tr}}\left( \sum_{j \neq k} A_{j,k}^* A_{j,k}\right)^m \notag\\
&\leq {\operatorname{Tr}}\left(\sum_{j,k}
A_{j,k}^* A_{j,k}\right)^m = {\operatorname{Tr}}\left( {s}n I_\Lambda\right)^m  = {s}^{m+1} n^m,  \notag\end{aligned}$$ where ${\operatorname{Tr}}$ denotes the trace. Furthermore, since $A_{j,k}^* = A_{k,j}$ we have $\sum_{j\neq k} A_{j,k}^* A_{j,k} =
\sum_{j\neq k} A_{j,k} A_{j,k}^*$. Let $F$ denote the block matrix $F = (\tilde{A}_{j,k})_{j,k}$ where $\tilde{A}_{j,k} = A_{j,k}$ if $j \neq k$ and $\tilde{A}_{j,j} = 0$. Using once again that the entries of all matrices are non-negative we obtain $$\begin{aligned}
& \|F\|_{S_{2m}}^{2m} = {\operatorname{Tr}}\left[(F^*F)^m\right] \notag\\
& = {\operatorname{Tr}}\left[\sum_{\substack{j_1,j_2,\hdots,j_m\\k_1,k_2,\hdots,k_m}} \tilde{A}_{j_1,k_1}^* \tilde{A}_{j_1,k_2} \tilde{A}_{j_2,k_2}^* \tilde{A}_{j_2,k_3} \cdots \tilde{A}^*_{j_m,k_m} \tilde{A}_{j_m,k_1}\right]\notag\\
& \leq {\operatorname{Tr}}\sum_{k_1,\hdots,k_m} \left[\sum_{j_1} A_{j_1,k_1}^* A_{j_1,k_2} \cdots \sum_{j_m} A_{j_m,k_m}^* A_{j_m,k_1}\right]\notag\\
& = {s}^m {\operatorname{Tr}}\sum_{k_1,\hdots,k_m} \left[R_\Lambda D_{k_1}^* P_\Omega D_{k_2} R_\Lambda^* R_\Lambda D_{k_2}^* P_\Omega D_{k_3} R_\Lambda^* \cdots
\right.
\notag\\
& \phantom{=  {s}^m {\operatorname{Tr}}\sum_{k_1,\hdots,k_m} [}\left.
\cdots R_\Lambda D_{k_m}^* P_\Omega D_{k_1} R_\Lambda^*\right],\notag\end{aligned}$$ where we applied also once more. Using the cyclicity of the trace and applying another time, together with the fact that $T_k = T_{-k}^*$ and $S_k = S_{N-k}^*$, gives $$\begin{aligned}
&\|F\|_{S_{2m}}^{2m} \leq {s}^m {\operatorname{Tr}}\left[\sum_{k_1} D_{k_1} P_\Lambda D_{k_1}^* P_\Omega \sum_{k_2} D_{k_2} P_\Lambda D_{k_2}^* P_\Omega \cdots
\right. \notag\\
& \phantom{\|F\|_{S_{2m}}^{2m} \leq }
\left. \cdots \sum_{k_n} D_{k_n} P_\Lambda D_{k_n}^* P_\Omega\right] = {s}^{2m} {\operatorname{Tr}}[ P_\Omega]
= n {s}^{2m}.\notag\end{aligned}$$ Since by assumption (\[cond:conditioning\]) ${s}\leq n$ it follows that $$\begin{aligned}
&\|F\|_{S_{2m}}^{2m} \leq \|(\sum_{j\neq k} A_{j,k}^* A_{j,k})^{1/2}\|_{S_{2m}}^{2m}\notag\\
& = \|(\sum_{j\neq k} A_{j,k} A_{j,k}^*)^{1/2}\|_{S_{2m}}^{2m} \,\leq\, n^m {s}^{m+1}.\notag\end{aligned}$$ Using $\|X_\Lambda\| = \|X_\Lambda\|_{S_\infty} \leq \|X_\Lambda\|_{S_p}$ and applying the Khintchine inequality in Theorem \[Bernoulli\_Khintchine\] we obtain for an integer $m$ $$\begin{aligned}
&{{\mathbb{E}}}\|X_\Lambda\|^{2m} = {{\mathbb{E}}}\|A_\Lambda^* A_\Lambda - I_\Lambda\|^{2m} \leq
{{\mathbb{E}}}\|A_\Lambda^* A_\Lambda - I_\Lambda\|_{S_{2m}}^{2m}\notag\\
&= \frac{1}{n^{2m}} {{\mathbb{E}}}\|\sum_{j\neq k} \epsilon_j \epsilon_k A_{j,k}\|_{S_{2m}}^{2m}\leq\, 2 (2\pi)^{2m} \left(\frac{(2m)!}{2^m m!}\right)^{2} \frac{{s}^{m+1}}{n^m}\notag.\end{aligned}$$ Stirling’s formula gives $$\label{Stirling1}
\frac{(2m)!}{2^m m!} = \frac{\sqrt{2\pi 2m} (2m/e)^{2m} e^{\lambda_{2m}}}{2^m \sqrt{2\pi m}
(m/e)^m e^{\lambda_m}}
\leq \sqrt{2}\, (2/e)^m m^m,$$ where $\frac{1}{12m+1} \leq \lambda_m \leq \frac{1}{12m}$. An application of Hölder’s inequality yields for $\theta \in [0,1]$ and an arbitrary random variable $Z$. $$\begin{aligned}
{{\mathbb{E}}}|Z|^{2m+2\theta} & =
{{\mathbb{E}}}[|Z|^{(1-\theta)2m} |Z|^{\theta (2m+2)}]\notag\\
& \leq ({{\mathbb{E}}}|Z|^{2m})^{1-\theta} ({{\mathbb{E}}}|Z|^{2m+2})^\theta.\label{interpol1}\end{aligned}$$ Combining our estimates above gives $$\begin{aligned}
&{{\mathbb{E}}}\|X_\Lambda\|^{2m+2\theta}
\leq ({{\mathbb{E}}}\|X_\Lambda\|^{2m})^{1-\theta} ({{\mathbb{E}}}\|X_\Lambda\|^{2m+2})^{\theta}\notag\\
&\leq 4 (2\pi)^{2m+2\theta} (2/e)^{2m+2\theta} m^{2m(1-\theta)} (m+1)^{2\theta(m+1)} \frac{{s}^{m+\theta+1}}{n^{m+\theta}}\notag\\
&\leq 4 \left(\frac{4\pi}{e}\right)^{2m+2\theta} (m+\theta)^{2m+2\theta}
 \frac{{s}^{m+\theta+1}}{n^{m+\theta}},\notag\end{aligned}$$ where we used the inequality between the geometric and arithmetic mean in the third step. In other words, for $p\geq 2$, $$\left({{\mathbb{E}}}\|X_\Lambda\|^{p}\right)^{1/p} \leq \frac{2\pi}{e} \sqrt{\frac{{s}}{n}} (4{s})^{1/p} p.$$ An application of Lemma \[lem\_moment1\] with the optimal value $\kappa = 1$ yields $${{\mathbb{P}}}\left(\|X_\Lambda\| \geq 2\pi \sqrt{\frac{{s}}{n}} u\right)
\leq 4{s}e^{-u}$$ for all $u\geq 2$. Setting the right hand side equal $\epsilon$ shows that $\|X_\Lambda\| \leq \delta$ with probability at least $1-\epsilon$ provided $$n \geq (2\pi)^2 \delta^{-2} s \log^2(4{s}/\epsilon).$$ This completes the proof of Theorem \[thm:condition\].

Acknowledgements {#acknowledgements .unnumbered}
================

This work was supported by the Hausdorff Center for Mathematics, University of Bonn and the WWTF project SPORTS (MA 07-004).

[10]{} \[1\][\#1]{} url@samestyle \[2\][\#2]{} \[2\][[l@\#1=l@\#1\#2]{}]{}

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Flow tube studies of the C((3)P) reactions with ethylene and propylene.
Product detection studies of C((3)P) atom reactions with ethylene, C2H4(X(1)Ag) and propylene, C3H6(X(1)A') are carried out in a flow tube reactor at 332 K and 4 Torr (553.3 Pa) under multiple collision conditions. Ground state carbon atoms are generated by 193 nm laser photolysis of carbon suboxide, C3O2 in a buffer of helium. Thermalized reaction products are detected using tunable VUV photoionization and time of flight mass spectrometry. For C((3)P) + ethylene, propargyl (C3H3) is detected as the only molecular product in agreement with previous studies on this reaction. The temporal profiles of the detected ions are used to discriminate C((3)P) reaction products from side reaction products. For C((3)P) + propylene, two reaction channels are identified through the detection of methyl (CH3) and propargyl (C3H3) radicals for the first channel and C4H5 for the second one. Franck-Condon Factor simulations are employed to infer the C4H5-isomer distribution. The measured 1 : 4 ratio for the i-C4H5 isomer relative to the methylpropargyl isomers is similar to the C4H5 isomer distribution observed in low-pressure flames and differs from crossed molecular beams data. The accuracy of these isomer distributions is discussed in view of large uncertainties on the photoionization spectra of the pure C4H5 isomers.


It's also possible to state the time in Italian by saying the hour followed by the minutes, followed, if necessary, by the words di mattina (from 5am until noon), di pomeriggio (from noon until 6pm), di sera (from 6pm until midnight), or di notte (from midnight until 5am), eg:


1. Field of the Invention
The present invention relates to a multiflavor beverage dispenser including a flow-rate control valve assembly and a microprocessor control system for controlling the ratio of diluent to concentrate of a post-mix beverage.
2. Description of Background Art
Multiflavor beverage dispensers, including microprocessors for providing ratio control of diluent to concentrate of post-mix beverage are generally known. An example of such a multiflavor beverage dispenser is disclosed in U.S. Pat. No. 4,487,333, issued Dec. 11, 1984 to Pounder et al.
Although the system of Pounder functions quite well for its intended purposes, a need in the art exists for a more compact apparatus with improved accuracy for ratio control, especially under conditions where the variation in system performance keeps the desired flow rate of the post-mix beverage from being reached.


Q:

How much JVM memory java heap space does the ExtJS Application uses

I would like to know how much of the JMV min memory and max memory (java heap space) that i need to set for the ExtJS application to perform well on my production server which is JBOSS server.
Is there a configuration file that is created after i build the application where i can set min, max memory of JVM to be used.
Can i test how much memory does ExtJS application uses.
Can i dictate server to use (i.e. with help of some configuration file) or allot a specific amount of JVM memory for my ExtJS Application.
I would like to know it so that i can set the min, max JVM options on my production server.
There is configuration file that talks about the JVM memory, but it is the memory that is being used when we build the application. The link that talks about it is
http://docs.sencha.com/extjs/4.1.3/#!/guide/command_advanced
Any kind of help is appreciated, Thanks.

A:

ExtJS is a JavaScript GUI Framework which runs on the clients and not the server side.


Anaheim Mayor Addresses Conflict of Interest Issues

By |October 12, 2011

Anaheim Mayor Tom Tait has addressed the resignation of building division head Scott Fazekas, who had been under fire since revelation that his firm had been getting the lion’s share of the city’s outsourced plan check review work.

“We have to be concerned about perceptions of conflict,” Tait said Oct. 11 when asked about how outsourced work should be handled.

Tait said that as the co-owner of two consulting firms, he wouldn’t allow the city to contract with his companies. “For me, as mayor, it’s not OK to have a contract with the city of Anaheim,” Tait said.

He would not comment, however, on whether that standard extends to other public officials.

Fazekas resigned last week in the aftermath of a Voice of OC article that showed he had received the majority of the division’s outsourced plan review work during his tenure.

Fazekas’ company billed the city for at least $18,954 in building plan review work between his appointment late in 2009 and June of this year, according to city records. During that period, Fazekas’ firm received 80 percent of the city’s outsourced plan review work.

A clause in Fazekas’ contract bars him from being involved in any outsourcing decisions. He said others in the building division made the decisions to send plan check work to his firm.

“I wanted to make sure there wasn’t a perception of exactly what you’re talking about,” Fazekas said previously. “We specified that I would have no involvement.”

Tracy Westen, CEO of the Los Angeles-based Center for Governmental Studies, questioned that relationship between Fazekas and staffers working in the building division.

“That doesn’t strike me as a healthy relationship,” Westen said. “If a person’s working for you, they may be under some explicit or implicit psychological pressure to send contracts your way.”


/*********************************************************
 * From C PROGRAMMING: A MODERN APPROACH, Second Edition *
 * By K. N. King                                         *
 * Copyright (c) 2008, 1996 W. W. Norton & Company, Inc. *
 * All rights reserved.                                  *
 * This program may be freely distributed for class use, *
 * provided that this copyright notice is retained.      *
 *********************************************************/

/* justify.c (Chapter 15, page 363) */
/* Formats a file of text */

#include <string.h>
#include "line.h"
#include "word.h"

#define MAX_WORD_LEN 20

int main(void)
{
  char word[MAX_WORD_LEN+2];
  int word_len;

  clear_line();
  for (;;) {
    read_word(word, MAX_WORD_LEN+1);
    word_len = strlen(word);
    if (word_len == 0) {
      flush_line();
      return 0;
    }
    if (word_len > MAX_WORD_LEN)
      word[MAX_WORD_LEN] = '*';
    if (word_len + 1 > space_remaining()) {
      write_line();
      clear_line();
    }
    add_word(word);
  }
}


Description

Amazing MOUNTAIN VIEW minutes west of Eagle Hill COOP. Great opportunity to choose which lot to build your DREAM HOME on in a central to Olds, Sundre & Bowden location. Paved access to internal cul de sac where these lots overlook a great valley view to the west with Electricity to lot line. Developer to approve home idea and plans- no mobiles and potential for seller financing. Come see the views today!

Description

Amazing MOUNTAIN VIEW minutes west of Eagle Hill COOP. Great opportunity to choose which lot to build your DREAM HOME on in a central to Olds, Sundre & Bowden location. Paved access to internal cul de sac where these lots overlook a great valley view to the west with Electricity to lot line. Developer to approve home idea and plans- no mobiles and potential for seller financing. Come see the views today!

Property Features

Monthly Payment Calculator

Related Properties

Data supplied by CREB®’s MLS® System. CREB® is the owner of the copyright in its MLS® System. The Listing data is deemed reliable but is not guaranteed accurate by CREB®. The information provided is for consumers’ personal, non-commercial use and may not be used for any purpose other than to identify prospective properties consumers may be interested in purchasing.
The trademarks MLS®, Multiple Listing Service® and the associated logos are owned by The Canadian Real Estate Association (CREA) and identify the quality of services provided by real estate professionals who are members of CREA. Used under license. The trademarks REALTOR®, REALTORS®, and the REALTOR® logo are controlled by The Canadian Real Estate Association (CREA) and identify real estate professionals who are members of CREA. Used under license. Data last updated: 2019-02-22T09:29:33.97.

www.livelovecalgary.com/homes/73714549

3
4141 Twp Rd 340
Rural Mountain View County,
AB
T0M 1X0

Price: $145,000

Status: Active

Updated: 46 min ago

ID#: C4123342

0

Beds

0

Baths

0

½ Baths

2

Acres

Neighbourhood:

None

Area:

Mountain View

School District:

Chinooks Edge

Property Description

Amazing MOUNTAIN VIEW minutes west of Eagle Hill COOP. Great opportunity to choose which lot to build your DREAM HOME on in a central to Olds, Sundre & Bowden location. Paved access to internal cul de sac where these lots overlook a great valley view to the west with Electricity to lot line. Developer to approve home idea and plans- no mobiles and potential for seller financing. Come see the views today!

Exterior Features

Garage Yn
No
Lot Shape
Irregular
Site Influences
ViewView Mountain

Property Features

Data supplied by CREB®’s MLS® System. CREB® is the owner of the copyright in its MLS® System. The Listing data is deemed reliable but is not guaranteed accurate by CREB®. The information provided is for consumers’ personal, non-commercial use and may not be used for any purpose other than to identify prospective properties consumers may be interested in purchasing.
The trademarks MLS®, Multiple Listing Service® and the associated logos are owned by The Canadian Real Estate Association (CREA) and identify the quality of services provided by real estate professionals who are members of CREA. Used under license. The trademarks REALTOR®, REALTORS®, and the REALTOR® logo are controlled by The Canadian Real Estate Association (CREA) and identify real estate professionals who are members of CREA. Used under license. Data last updated: 2019-02-22T09:29:33.97.

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About Us

Data supplied by CREB®’s MLS® System. CREB® is the owner of the copyright in its MLS® System. The Listing data is deemed reliable but is not guaranteed accurate by CREB®. The information provided is for consumers’ personal, non-commercial use and may not be used for any purpose other than to identify prospective properties consumers may be interested in purchasing.
The trademarks MLS®, Multiple Listing Service® and the associated logos are owned by The Canadian Real Estate Association (CREA) and identify the quality of services provided by real estate professionals who are members of CREA. Used under license. The trademarks REALTOR®, REALTORS®, and the REALTOR® logo are controlled by The Canadian Real Estate Association (CREA) and identify real estate professionals who are members of CREA. Used under license. Data last updated: 2019-02-22T09:29:33.97.


1. Field of the Invention
The present disclosure relates to devices that facilitate the maintenance of fluid systems and in particular, to an adapter that connects to a filter for the purging of a line of a fluid system.
2. Description of the Related Art
Oil heating systems are one example of a common fluid system that employs a pump that draws a fluid from a tank through a feed or input line to the pump. An oil filter is located upstream in the input line from the pump that filters out undesirable particles from the oil. Problems occur in fluid systems, such as in this example when the input line upstream from the filter gradually becomes clogged with deposits from the oil. This is a sizable problem for oil systems because it reduces the flow of oil through the input line. This strains the pump and reduces the efficiency of the overall system. As a result of this problem, oil lines are routinely purged as part of a maintenance program.
The purging of oil lines is a burdensome process. For example, in order to purge the line from the oil tank to the filter of an oil burning home heating system a technician closes a first in-line valve in proximity to the oil tank and disconnects the line at a second in-line valve in proximity to the oil filter. A hand pneumatic push-pump is connected to the disconnected line, the push-pump is pressurized and the first in-line valve is opened. The pump supplies a flow of pressurized air into the input line that forces or purges the oil and oil deposits from the line into tank. This process is then reversed for the normal operation of the oil system. The disconnecting, purging and reconnecting process is excessively time consuming due to complications that include the first in-line valve in proximity to the oil tank being located external to the structure of the building, at a remote section inside the structure or barely accessible due to clutter surrounding the tank. Technicians have to make multiple trips between the remote first in-line valve in proximity to the oil tank and the broken line in proximity to the oil filter in order to perform this maintenance.
Heretofore there has never been a reliable device and/or method for the process of purging oil filter input lines without the time consuming process of breaking the line and the associated requirement for the repeated opening and closing of a remote shut off valve.


Effectiveness of resident-prepared conferences in teaching imaging utilization guidelines to radiology residents.
The authors evaluated the effectiveness of a resident-prepared conference series for teaching imaging utilization guidelines to radiology residents. Brown University radiology residents (n = 17) gave 61 presentations on imaging utilization to their colleagues during 16 1-hour conferences. The residents were later examined on the topics presented and surveyed about their familiarity with the American College of Radiology appropriateness criteria, their exposure to issues of cost-effectiveness, and their degree of confidence in providing imaging consultation. The same examination and survey were administered to control residents from the University of Wisconsin (n = 14) and the Oregon Health Sciences University (n = 14). Scores were compared by using linear regression and Wilcoxon rank sum tests. Controlling for years in radiology residency, residents at Brown scored on average 16.0% (standard error = 2.2%) higher than residents at the other universities (P < .001). Controlling for institution, 3rd- and 4th-year residents scored on average 7.4% (standard error = 2.1%) higher than 1st- and 2nd-year residents (P = .001). Brown residents expressed more familiarity with American College of Radiology appropriateness criteria and appeared to have more exposure to cost-effectiveness issues in conferences than residents at Wisconsin or Oregon Health Sciences University (P < .005). Residents from the three universities did not differ in their level of confidence in providing imaging consultation. Resident-prepared conferences are an effective means of teaching imaging utilization guidelines to residents, but they do not affect the residents' perception of their ability to provide imaging consultation.


#!/bin/sh

DEF_VER=v5.17_devel

# First try git describe (if running on a git repo),
# then use default version from above (for release tarballs).
if test -d .git -o -f .git
then
    # if a commandline argument was given, assume it was a ref that git can interpret
    if [ "$#" -ne 0 ]; then
        GIT_VN=$(git describe --match "v[0-9].[0-9]*" --always --long $*)
    else
        GIT_VN=$(git describe --match "v[0-9].[0-9]*" --dirty --always --long)
    fi
    if echo "$GIT_VN" | grep -Eq "^v[0-9].[0-9]"
    then
        VN="$GIT_VN"
    else
        VN="$DEF_VER"-none-g"$GIT_VN"
    fi
else
    VN="$DEF_VER"
fi

echo $VN