scholarly journals Differentiating convex functions constructively

2020 ◽  
Vol 12 ◽  
Author(s):  
Hannes Diener ◽  
Matthew Hendtlass
Keyword(s):  
Convex Functions ◽  
Classical Analysis ◽  
Almost Everywhere ◽  
Increasing Functions ◽  
Do So

In classical analysis, both  convex functions and  increasing functions \([0,1] \to \RR\) are differentiable almost everywhere. We will show that constructively, while we can prove this for convex functions, we cannot do so for increasing ones.

scholarly journals An Alexsandrov type theorem for k-convex functions

2005 ◽  
Vol 71 (2) ◽  
pp. 305-314 ◽  
Author(s):  
Nirmalendu Chaudhuri ◽  
Neil S. Trudinger
Keyword(s):  
Positive Integer ◽  
Convex Functions ◽  
Type Theorem ◽  
Classical Theorem ◽  
Almost Everywhere

In this note we show that k-convex functions on ℝn are twice differentiable almost everywhere for every positive integer k > n/2. This generalises Alexsandrov's classical theorem for convex functions.

Addition Theorems and Binary Expansions

1995 ◽  
Vol 47 (2) ◽  
pp. 262-273 ◽  
Author(s):  
Jonathan M. Borwein ◽  
Roland Girgensohn
Keyword(s):  
Unknown Function ◽  
Functional Equations ◽  
Elliptic Integral ◽  
Addition Theorems ◽  
Trigonometric Functions ◽  
Binary Expansion ◽  
Almost Everywhere ◽  
Image Position ◽  
Increasing Functions ◽  
Inverse Trigonometric Functions

AbstractLet an interval I ⊂ ℝ and subsets D0, D1 ⊂ I with D0 ∪ D1 = I and D0 ∩ D1 = Ø be given, as well as functions r0: D0 → I, r1: D1 → I. We investigate the system (S) of two functional equations for an unknown function f: I → [0, 1]: We derive conditions for the existence, continuity and monotonicity of a solution. It turns out that the binary expansion of a solution can be computed in a simple recursive way. This recursion is algebraic for, e.g., inverse trigonometric functions, but also for the elliptic integral of the first kind. Moreover, we use (S) to construct two kinds of peculiar functions: surjective functions whose intervals of constancy are residual in I, and strictly increasing functions whose derivative is 0 almost everywhere.

A characterization of almost everywhere convex functions

2001 ◽  
Vol 4 (1) ◽  
pp. 49-60 ◽  
Author(s):  
Davide La Torre ◽  
Matteo Rocca
Keyword(s):  
Convex Functions ◽  
Almost Everywhere

scholarly journals Riemann-Liouville Fractional integral operators with respect to increasing functions and strongly $ (\alpha, m) $-convex functions

2021 ◽  
Vol 6 (10) ◽  
pp. 11403-11424
Author(s):  
Ghulam Farid ◽  
◽  
Hafsa Yasmeen ◽  
Hijaz Ahmad ◽  
Chahn Yong Jung ◽  
...  
Keyword(s):  
Convex Functions ◽  
Fractional Integral ◽  
Integral Operators ◽  
Integral Inequalities ◽  
Fractional Integrals ◽  
Fractional Integral Operators ◽  
Strongly Convex ◽  
Increasing Functions ◽  
Integral Identities

<abstract><p>In this paper Hadamard type inequalities for strongly $ (\alpha, m) $-convex functions via generalized Riemann-Liouville fractional integrals are studied. These inequalities provide generalizations as well as refinements of several well known inequalities. The established results are further connected with fractional integral inequalities for Riemann-Liouville fractional integrals of convex, strongly convex and strongly $ m $-convex functions. By using two fractional integral identities some more Hadamard type inequalities are proved.</p></abstract>

The God That Won: Eugen Kogon and the Origins of Cold War Liberalism

2020 ◽  
Vol 55 (2) ◽  
pp. 339-363
Author(s):  
James Chappel
Keyword(s):  
Human Rights ◽  
Cold War ◽  
Liberal Democracy ◽  
West Germany ◽  
Concentration Camps ◽  
Important Phenomenon ◽  
Almost Everywhere ◽  
The Cold War ◽  
Do So

Eugen Kogon (1903–87) was one of the most important German intellectuals of the late 1940s. His writings on the concentration camps and on the nature of fascism were crucial to West Germany’s fledgling transition from dictatorship to democracy. Previous scholars of Kogon have focused on his leftist Catholicism, which differentiated him from the mainstream. This article takes a different approach, asking instead how Kogon, a recovering fascist himself, came to have so much in common with his peers in West Germany and in the Cold War West. By 1948, he fluently spoke the new language of Cold War liberalism, pondering how human rights and liberal democracy could be saved from totalitarianism. He did not do so, the article argues, because he had decided to abandon his principles and embrace a militarized anti-Communist cause. Instead, he transitioned to Cold War liberalism because it provided a congenial home for a deeply Catholic thinker, committed to a carceral understanding of Europe’s fascist past and a federalist vision for its future. The analysis helps us to see how European Catholics made the Cold War their own – an important phenomenon, given that Christian Democrats held power almost everywhere on the continent that was not controlled by Communists. The analysis reveals a different portrait of Cold War liberalism than we usually see: less a smokescreen for American interests, and more a vessel for emancipatory projects and ideals that was strategically employed by diverse actors across the globe.

scholarly journals Fractional integral inequalities of Hermite-Hadamard type for convex functions with respect to a monotone function

Filomat ◽  
2020 ◽  
Vol 34 (7) ◽  
pp. 2401-2411
Author(s):  
Pshtiwan Mohammed
Keyword(s):  
Convex Functions ◽  
Fractional Integral ◽  
Special Functions ◽  
Monotone Function ◽  
Type Inequality ◽  
Integral Inequalities ◽  
Fractional Integrals ◽  
Increasing Functions

In the literature, the left-side of Hermite-Hadamard?s inequality is called a midpoint type inequality. In this article, we obtain new integral inequalities of midpoint type for Riemann-Liouville fractional integrals of convex functions with respect to increasing functions. The resulting inequalities generalize some recent integral inequalities and Riemann-Liouville fractional integral inequalities established in earlier works. Finally, applications of our work are demonstrated via the known special functions.

ON THE PRINCIPLE OF CONTROLLED L∞ CONVERGENCE IMPLIES ALMOST EVERYWHERE CONVERGENCE FOR GRADIENTS

2007 ◽  
Vol 09 (01) ◽  
pp. 21-30
Author(s):  
KEWEI ZHANG
Keyword(s):  
Continuous Part ◽  
Absolutely Continuous ◽  
Almost Everywhere Convergence ◽  
Lipschitz Mappings ◽  
Open Set ◽  
Almost Everywhere ◽  
Increasing Functions

Let Ω ⊂ ℝn be a bounded open set. If a sequence fk : Ω → ℝN converges to f in L∞ in a certain "controlled" manner while bounded in W1,p (1 < p < + ∞) or BV, we show that f ∈ W1,p (respectively, f ∈ BV) and ∇fk → ∇f almost everywhere, where ∇fk and ∇f are the usual gradients if fk ∈ W1,p (respectively, the absolutely continuous part of the gradient measures if fk ∈ BV). Our main theorem generalizes results for Lipschitz mappings. We show by an example that when p = 1, the limit of a sequence of increasing functions may fail to be in W1,1 and can even be nowhere C1.

Puccinia menthae. [Descriptions of Fungi and Bacteria].

1964 ◽  
Author(s):  
G. F. Laundon
Keyword(s):  
Geographical Distribution ◽  
Spore Germination ◽  
Soil P ◽  
Almost Everywhere ◽  
Do So

Abstract A description is provided for Puccinia menthae. Information is included on the disease caused by the organism, its transmission, geographical distribution, and hosts. HOSTS: On species of Blephila, Bystropogon, Calamintha, Clinopodium, Cunila, Hedeoma, Hyssopus, Lycopus, Melissa, Mentha, Micromeria, Monarda, Monardella, Nepeta, Ocimum, Origanum, Pycnanthemum, Satureja, Thymus and Ziziphora. DISEASE: Mint rust is associated with two distinct symptoms: systemically infected swollen shoots with elongated chlorotic internodes bearing malformed chlorotic leaves; and localized lesions on leaves following the development of uredia and telia, which when abundant cause necrosis of large areas and premature defoliation. GEOGRAPHICAL DISTRIBUTION: Chiefly North Temperate but occurs almost everywhere mint is grown (CMI Map 211). TRANSMISSION: Probably by wind and contaminated rhizomes. For details of spore germination and infection see 9: 558 and 15: 527. There is some doubt as to whether P. menthae is truly systemic: some authors have reported mycelium in the rhizome tissues (Ogilvie and Hickman, 17: 6) but others have failed to do so. The ease with which the rhizomes can be cleared of the disease suggests that reinfection of the young stems probably takes place every year afresh from spores adhering to the rhizomes and mixed in the soil.

scholarly journals A class of null sets associated with convex functions on Banach spaces

1990 ◽  
Vol 42 (2) ◽  
pp. 315-322 ◽  
Author(s):  
John Rainwater
Keyword(s):  
Large Class ◽  
Banach Spaces ◽  
Convex Functions ◽  
Measure Zero ◽  
Almost Everywhere

A generalisation of the notion of “sets of measure zero” for arbitrary Banach spaces is defined so that continuous convex functions are automatically Gateaux differentiable “almost everywhere”. It is then shown that this class of sets satisfies all the properties tht one expects of sets of measure zero. Moreover (in a certain large class of Banach spaces, at least) nonempty open sets are not of “measure zero”.

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