Search

May's classical theorem states that in a single-winner choose-one voting system with just two candidates, majority rule is the only social choice function satisfying anonimity, neutrality and positive responsiveness axiom. Anonimity and neutrality are usually regarded as very natural constraints on the social choice function. Positive responsiveness, on the other hand, is sometimes deemed too strong of an axiom, which stimulates further search for less stringent conditions. One viable substitute is Gerhard J. Woeginger's "reducibility to subsocieties". We demonstrate that the condition generalizes to more than two candidates and, consequently, characterizes majority rule for elections with multiple candidates.

It is common knowledge that the Fourier transform enjoys the convolution property, i.e., it turns convolution in the time domain into multiplication in the frequency domain. It is probably less known that this property characterizes the Fourier transform amongst all linear and bounded operators $T:L^1 \longrightarrow C^b.$ Thus, a natural question arises: are there other features characterizing Fourier transform besides the convolution property? We answer this query in the affirmative by investigating the time differentiation property and its discrete counterpart, used to characterize discrete-time Fourier transform. Next, we move on to locally compact abelian groups, where differentiation becomes meaningless, but the Fourier transform can be characterized via time shifts. The penultimate section of the paper returns to the convolution characterization, this time in the context of compact (not necessarily abelian) groups. We demonstrate that the proof existing in the literature can be greatly simplified. Lastly, we hint at the possibility of other transforms being characterized by their properties and demonstrate that the Hankel transform may be characterized by a Bessel-type differential property.

Inspired by Jaming's characterization of the Fourier transform on specific groups via the convolution property, we provide a novel approach which characterizes the Fourier transform on any locally compact abelian group. In particular, our characterization encompasses Jaming's results. Furthermore, we demonstrate that the cosine transform as well as the Laplace transform can also be characterized via a suitable convolution property.

The primary motivation behind this paper is an attempt to provide a thorough explanation of how the Mellin transform arises naturally in a process akin to the construction of the celebrated Gelfand transform. We commence with a study of a class of Schwartz functions $\mathcal{S}(\mathbb{R}_+),$ where $\mathbb{R}_+$ is the set of all positive real numbers. Various properties of this Fr\'echet space are established and what follows is an introduction of the Mellin convolution operator, which turns $\mathcal{S}(\mathbb{R}_+)$ into a commutative Fr\'echet algebra. We provide a simple proof of Mellin-Young convolution inequality and go on to prove that the structure space $\Delta(\mathcal{S}(\mathbb{R}_+),\star)$ (the space of nonzero, linear, continuous and multiplicative functionals $m:\mathcal{S}(\mathbb{R}_+)\longrightarrow \mathbb{R}$) is homeomorphic to $\mathbb{R}.$ Finally, we show that the Mellin transform arises in a process which bears a striking resemblance to the construction of the Gelfand transform.

The aim of the paper is to compare different approximation algorithms for the travelling salesperson problem. We pick the most popular and widespread methods known in the literature and contrast them with a novel approach (the polygonal Christofides algorithm) described in our previous work. The paper contains a brief summary of theory behind the algorithms and culminates in a series of numerical simulations (or "experiments"), whose purpose is to determine "the best" approximation algorithm for the travelling salesperson problem., Comment: arXiv admin note: text overlap with arXiv:2105.07275

The goal of the paper is to provide a detailed explanation on how the (continuous) cosine transform and the discrete(-time) cosine transform arise naturally as certain manifestations of the celebrated Gelfand transform. We begin with the introduction of the cosine convolution $\star_c$, which can be viewed as an "arithmetic mean" of the classical convolution and its "twin brother", the anticonvolution. The d'Alambert property of $\star_c$ plays a pivotal role in establishing the bijection between $\Delta(L^1(G),\star_c)$ and the cosine class $\mathcal{COS}(G),$ which turns out to be an open map if $\mathcal{COS}(G)$ is equipped with the topology of uniform convergence on compacta $\tau_{ucc}$. Subsequently, if $G = \mathbb{R},\mathbb{Z}, S^1$ or $\mathbb{Z}_n$ we find a relatively simple topological space which is homeomorphic to $\Delta(L^1(G),\star_c).$ Finally, we witness the "reduction" of the Gelfand transform to the aforementioned cosine transforms.

With the aid of the relaxed polygonal inequality (introduced by Fagin et al.) we strive to extend the applicability of Christofides approximation technique to the scope of all complete finite weighted graphs with positive weights. First section acquaints the Reader with the class of semimetric graphs and proves that every finite graph admits $\gamma$-polygon structure. Sections 2 and 3 establish the necessary notions from the graph and optimization theory to tackle the Traveling Salesperson Problem. In section 4 the minimal spanning tree method is introduced, while section 5 focuses on the analysis of this method through the lens of $\gamma$-polygon graphs. The final section of the paper adjusts the technique of Christofides by obtaining $\frac{3\gamma}{2}-$approximation for the TSP.

In the paper we make an effort to answer the question ``What if Arzel\`a and Ascoli lived long enough to see Pego theorem?''. Giulio Ascoli and Cesare Arzel\`a died in 1896 and 1912, respectively, so they could not appreciate the characterization of compact families in $L^2(\mathbb{R}^N)$ provided by Robert L. Pego in 1985. Unlike the Italian mathematicians, Pego employed various tools from harmonic analysis in his work (for instance the Fourier transform or the Hausdorff-Young inequality). Our article is meant to serve as a bridge between Arzel\`a-Ascoli theorem and Pego theorem (for $L^1(G)$ rather than $L^2(G)$, $G$ being a locally compact abelian group). In a sense, the former is the ``raison d'\^etre'' of the latter, as we shall painstakingly demonstrate.

The primary aim of the paper is the study of Sobolev spaces in the context of Gelfand pairs. The article commences with providing a historical overview and motivation for the researched subject together with a summary of the current state of the literature. What follows is a general outline of harmonic analysis on Gelfand pairs, starting with a concept of positive-semidefinite functions, through spherical functions and spherical transform and concluding with the Hausdorff-Young inequality. The main part of the paper introduces the notion of Sobolev spaces on Gelfand pairs and studies the properties of these spaces. It turns out that Sobolev embedding theorems and Rellich-Kondrachov theorem still hold true in this generalized context (if certain technical caveats are taken into consideration).

In 1985, Robert L. Pego characterized compact families in $L^2(\reals)$ in terms of the Fourier transform. It took nearly 30 years to realize that Pego's result can be proved in a wider setting of locally compact abelian groups (works of G\'orka and Kostrzewa). In the current paper, we argue that the Fourier transform is not the only integral transform that is efficient in characterizing compact families and suggest the Laplace transform as a possible alternative.

In recent years, a lot of research was devoted to Simpson's rule for numerical integration. In the paper we study a natural successor of Simpson's rule, namely the Boole's rule. It is the Newton-Cotes formula in the case where the interval of integration is divided into four subintervals of equal length. With computer software assistance, we prove novel error bounds for Boole's rule.

In the paper, we study the images of circles, lines, balls and half-planes under Mobius transformations.

In the paper we look for an elegant proof of the characterization of relatively compact families in $L^p-$spaces. At first glance, the suggested approach may seem convoluted and lengthy, but we spare no effort to argue that our proof is in fact more natural than the ones existent in the literature. The key idea is that the three properties which characterize relative compactness in $L^p-$spaces ($L^p-$boundedness, $L^p-$equicontinuity and $L^p-$equivanishing) are "preserved" (or rather "inherited") when the family $\Ffamily\subset L^p$ is convolved with a continuous and compactly supported function. This new family turns out to be relatively compact in $C_0-$space and it remains to be demonstrated that relative compactness in $C_0-$space implies relative compactness of the original family $\Ffamily.$

In this note we consider the classical variational tools like: Ekelenad's Variational Principle, Mountain Pass Lemma and some of their corollaries subject to a parameter. Next, we investigate the behaviour of critical points obtained once a sequence of parameters is allowed to be convergent. Applications for the Dirichlet Boundary Value Problem on the Sierpi\'{n}ski Gasket are given in presence of assumptions which lead to fulfillment of the mountain geometry.

Our paper begins with a revision of spectral theory for commutative Banach algebras, which enables us to prove the $L^p_{\omega}-$conjecture for locally compact abelian groups. We follow an alternative approach to the one known in the literature. In particular, we do not resort to any structural theorems for locally compact groups. Subsequently, we discuss nilpotent, locally compact groups. The climax of the paper is the proof of the $L^p_{\omega}-$conjecture for these groups.

In the paper, we introduce the concept of weight with reasonable growth on a locally compact group $G$. We verify that these weights form a natural class to work with, by examining the most common examples. We proceed with the discussion of the $L^p-$conjecture. With the use of Riesz-Thorin-Stein-Weiss interpolation, we establish that \mbox{$L^p_{\omega}(G)\star L^p_{\omega}(G)\subset L^p_{\omega}(G)$}, $p>1$ implies that $L^p_{\omega}(G)\star L^q_{\omega}(G)\subset L^q_{\omega}(G)$ for $q$ which lies between $p$ and $p'$. At last, we confirm the $L^p_{\omega}-$conjecture for weights of $(p,q)-$reasonable growth., Comment: Error in the proof

As indicated in the title, this paper was inspired by Smithson's work on Arzel\`a-Ascoli theorem. Up to this point, the author's interest in the topic has led him to generalize Arzel\`a-Ascoli theorem in the context of uniform spaces. This approach was primarily influenced by J.L. Kelley's \textit{General topology}. Soon after, the author learnt that in 1971, R.E. Smithson carried over Kelley's approach to the case of point-compact multifunctions (rather than single-valued functions). Similarly, in this paper the author would like to carry over the result, presented in his previous paper \textit{Arzel\`a-Ascoli theorem in uniform spaces}, to the case of point-compact multifunctions.

The paper introduces the concept of quasimeasure of noncompactness. Motivated by the Arzel\`a-Ascoli theorem for $C^b(X,E)$, where $X$ is an Euclidean space and $E$ an arbitrary Banach space, we construct a quasimeasure for this space and study its properties. An analogon for Darbo fixed point theorem is obtained with the additional aid of measure of nonconvexity. The paper ends with possible application in integral equations. We prove that a Hammerstein operator with Carath\'{e}odory kernel and nonlinearity of a certain type has a fixed point.

In the paper, we generalize the Arzel\`a-Ascoli theorem in the setting of uniform spaces. At first, we recall well-known facts and theorems coming from monographs of Kelley and Willard. The main part of the paper introduces the notion of extension property which, similarly as equicontinuity, equates different topologies on $C(X,Y)$. This property enables us to prove the Arzel\`a-Ascoli theorem for uniform convergence. The paper culminates with applications, which are motivated by Schwartz's distribution theory. Using the Banach-Alaoglu-Bourbaki theorem, we establish relative compactness of subfamily of $C(\mathbb{R},\mathcal{D}'(\mathbb{R}^n))$.

In the paper, we recall the Wallman compactification of a Tychonoff space $T$ (denoted by $\text{Wall}(T)$) and the contribution made by Gillman and Jerison. Motivated by the Gelfand-Naimark theorem, we investigate the homeomorphism between $BC(T,\mathbb{R})$ and $BC(\text{Wall}(T),\mathbb{R})$. Along the way, we attempt to justify the advantages of Wallman compactification over other manifestations of Stone-\v{C}ech compactification. The main result of the paper is a new form of Arzel\`a-Ascoli theorem, which introduces the concept of equicontinuity along $\omega$-ultrafilters.

A version of Arzel\`a-Ascoli theorem for $X$ being $\sigma$-locally compact Hausdorff space is proved. The result is used in proving compactness of Fredholm, Hammerstein and Urysohn operators. Two fixed point theorems, for Hammerstein and Urysohn operator, are derived on the basis of Schauder theorem.