Discrete Arzelà-Ascoli Theorem on the Half Line and Its Application

The compactness criterion in l_◇={x(·)∈l~∞(0,∞))|lim t→∞ x(t)=x(∞)} is given and as an application,we study the existence of positive solutions of second order boundary value problems of difference equation on the half line by the fixedpoint theorem in cones.     

Beweis eines Satzes von Arzelà

Ohne Zusammenfassung     

A net and open-filter process of compactification and the Stone-?ech, Wallman compactifications

By a characterization of compact spaces in Section 1, a process of obtaining a compactification (X^∗,k) of an arbitrary topological space X is described in Section 2 by a combined approach of nets and open filters. The Wallman compactification can be embedded in X^∗ if X is Hausdorff and by a little modification, the compactification of X is the Stone-?ech compactification of X if X is Tychonoff.     

The Hyers theorem via the Markov–Kakutani fixed point theorem

We present an application of the Markov–Kakutani common fixed point theorem to the theory of stability of functional equation by proving some version of the Hyers theorem concerning approximate homomorphisms.     

A remark about the statistical Cesàro summability and the Orlicz-Pettis theorem

We prove a version of the Orlicz-Pettìs theorem within the frame of the Statistical Cesàro summability.     

The Rohde–Schramm theorem via the Gaussian free field

The Rohde–Schramm theorem states that Schramm–Loewner Evolution with parameter κ (or SLE_κ for short) exists as a random curve, almost surely, if κ ≠ 8. Here we give a new and concise proof of the result, based on the Liouville quantum gravity coupling (or reverse coupling) with a Gaussian free field. This transforms the problem of estimating the derivative of the Loewner flow into estimating certain correlated Gaussian free fields. While the correlation between these fields is not easy to understand, a surprisingly simple argument allows us to recover a derivative exponent first obtained by Rohde and Schramm [14], subsequently shown to be optimal by Lawler and Viklund [17], which then implies the Rohde–Schramm theorem.     

A theorem on the absolute Cesàro summability of factored fourier series

Summary In this paper, the author proves a theorem on the absoluteCesàro summability of factoredFourier series. His theorem extends a theorem ofMatsumoto and generalizes a theorem ofPrasad andBhatt.     

The Sandwich theorem via Pataraia’s fixed point theorem

Positivity - The Sandwich theorem (König in Archiv der Mathematik 23:500–1972, 1972) yields the existence of a linear functional sandwiched in between a given superlinear functional and...     

C. Arzelàs Abhandlung: Sulle serie di funzioni,parte prima

Ohne Zusammenfassung

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C. Arzelàs Abhandlung: Sulle serie di funzioni, parte prima

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Memorie della r. Accademia di Bologna ser 5. T. VIII, p. 131–186. Die Abschnitte I und II s. M.-H. S. 54 ff.     

C. Arzelàs Abhandlung: Sulle serie di funzioni,parte prima

Ohne Zusammenfassung

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C. Arzelàs Abhandlung: Sulle serie di funzioni, parte prima

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Memorie della r. Accademia di Bologna ser 5. T. VIII, p. 131–186.

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Diese Bearbeitung gibt von den drei Abschnitten, in welche die genannte Abhandlung zerfällt, zwei wieder. An der Vollendung des dritten wurde der begabte Verfasser durch die Erkrankung verhindert, welche leider seinen frühzeitigen Tod am 25. Oktober 1903 in seiner Heimat Ötz herbeiführte. Die Bearbeitung des dritten Abschnittes, die Herr Hofrat Stolz dem Unterzeichneten zu übertragen die Güte hatte, wird bald folgen.     

Hermite’s theorem via Galois cohomology

Archiv der Mathematik - An 1861 theorem of Hermite asserts that for every field extension E / F of degree 5 there exists an element of E whose minimal polynomial over F is of the...     

Cone theorem via Deligne–Mumford stacks

We prove the cone theorem for varieties with LCIQ singularities using deformation theory of stable maps into Deligne–Mumford stacks. We also obtain a sharper bound on −(K _X + D)-degree of (K _X + D)-negative extremal rays for projective -factorial log terminal threefold pairs (X, D).     

A Carathéodory theorem for the bidisk via Hilbert space methods

If φ is an analytic function bounded by 1 on the bidisk \({{\mathbb D}^2}\) and \({\tau\in\partial({\mathbb D}^2)}\) is a point at which φ has an angular gradient \({\nabla\varphi(\tau)}\) then \({\nabla\varphi(\lambda) \to \nabla\varphi(\tau)}\) as λ → τ nontangentially in \({{\mathbb D}^2}\). This is an analog for the bidisk of a classical theorem of Carathéodory for the disk. For φ as above, if \({\tau\in\partial({\mathbb D}^2)}\) is such that the lim inf of \({(1-|\varphi(\lambda)|)/(1-\|\lambda\|)}\) as λ → τ is finite then the directional derivative D _?δ φ(τ) exists for all appropriate directions \({\delta\in{\mathbb C}^2}\). Moreover, one can associate with φ and τ an analytic function h in the Pick class such that the value of the directional derivative can be expressed in terms of h.     

A Tauberian theorem for Cesàro summability of integrals

In this paper we give a proof of the generalized Littlewood Tauberian theorem for Cesàro summability of improper integrals.     

Quasiperiodic solutions of higher dimensional Duffing’s equations via the KAM theorem

In this paper, we consider the higher dimensional second order differential equations of the formẍ + ∇F(x,t) = 0,x ∈R ⁿ with a class of weakly coupled potentials F( x, t ), periodically depending on t. We prove the existence of infinitely many quasi-periodic solutions for such equations via the KAM theorem.     

Komlós's tiling theorem via graphon covers

Komlós [Komlós: Tiling Turán Theorems, Combinatorica, 2000] determined the asymptotically optimal minimum-degree condition for covering a given proportion of vertices of a host graph by vertex-disjoint copies of a fixed graph H, thus essentially extending the Hajnal–Szemerédi theorem that deals with the case when H is a clique. We give a proof of a graphon version of Komlós's theorem. To prove this graphon version, and also to deduce from it the original statement about finite graphs, we use the machinery introduced in [Hladký, Hu, Piguet: Tilings in graphons, arXiv:1606.03113]. We further prove a stability version of Komlós's theorem.     

A categorical property of the Stone-Čech compactification

We study some categorical properties of the functor O _β of weakly additive functionals acting in the category Tych of the Tychonoff spaces and their continuous mappings. We show that O _β preserves the weight of infinite-dimensional Tychonoff spaces, the singleton, and the empty set, and that O _β is monomorphic and continuous in the sense of T. Banakh, takes every perfect mapping to an epimorphism, and preserves intersections of functionally closed sets in a Tychonoff space.     

Orlicz-Pettis Type Theorems via Strong p-Cesàro Convergence

We obtain a new version of the Orlicz-Pettis theorem within the frame of the strong p-Cesàro convergence.     

On the Epireflectivity of the Wallman–Shanin-Type Compactification for Approach Spaces

In this paper, we study the functorial behaviour of the Wallman–Shanin-type compactification for weakly symmetric T ₁ approach spaces, as defined in (R. Lowen and M. Sioen [13]). We do this by generalizing the technique used by H. L. Bentley and S. A. Naimpally in their paper [2], yielding a recharacterization of our quantified compactification theory in terms of contiguity clusters of [0, ]-valued functionals and we also define a construct WS (which is the counterpart of SEP in [2]) on a suitable non-full subconstruct of which the Wallman–Shanin-type compactification determines an epireflection.