Proof Of Existence Research Articles

Computing Pareto-Optimal and Almost Envy-Free Allocations of Indivisible Goods

We study the problem of fair and efficient allocation of a set of indivisible goods to agents with additive valuations using the popular fairness notions of envy-freeness up to one good (EF1) and equitability up to one good (EQ1) in conjunction with Pareto-optimality (PO). There exists a pseudo-polynomial time algorithm to compute an EF1+PO allocation and a non-constructive proof of the existence of allocations that are both EF1 and fractionally Pareto-optimal (fPO), which is a stronger notion than PO. We present a pseudopolynomial time algorithm to compute an EF1+fPO allocation, thereby improving the earlier results. Our techniques also enable us to show that an EQ1+fPO allocation always exists when the values are positive and that it can be computed in pseudo-polynomial time. We also consider the class of k-ary instances where k is a constant, i.e., each agent has at most k different values for the goods. For such instances, we show that an EF1+fPO allocation can be computed in strongly polynomial time. When all values are positive, we show that an EQ1+fPO allocation for such instances can be computed in strongly polynomial time. Next, we consider instances where the number of agents is constant and show that an EF1+PO (likewise, an EQ1+PO) allocation can be computed in polynomial time. These results significantly extend the polynomial-time computability beyond the known cases of binary or identical valuations. We also design a polynomial-time algorithm that computes a Nash welfare maximizing allocation when there are constantly many agents with constant many different values for the goods. Finally, on the complexity side, we show that the problem of computing an EF1+fPO allocation lies in the complexity class PLS.

Propositional Type Theory of Indeterminacy

AbstractThe aim of this paper is to define a partial Propositional Type Theory. Our system is partial in a double sense: the hierarchy of (propositional) types contains partial functions and some expressions of the language, including formulas, may be undefined. The specific interpretation we give to the undefined value is that of Kleene’s strong logic of indeterminacy. We present a semantics for the new system and prove that every element of any domain of the hierarchy has a name in the object language. Finally, we provide a proof system and a (constructive) proof of completeness.

Allah Yang Menderita sebagai Kekuatan Menghadapi Penderitaan

This paper describes the suffering God as strength in facing suffering. This research departs from the philosophical-theological issues about evil and suffering in relation to the existence of God; If God exists why is there evil and suffering? Philosophy by using pure reason tries to keep looking for answers to the problem and the final answer is that there is no satisfactory answer. Whereas theology, based on revelation and with the help of reason, provides the final answer that closes the entire series of questions, namely that evil and suffering are mysteries. Humans will not get a definite answer to this problem, but one thing that is certain is that humans experience evil and suffering in the world. Therefore, there are two main problems involved in this research. First, what is the philosophical-theological view of the existence of God in relation to the phenomenon of evil and human suffering? Second, what is the Christian response to human suffering in light of the event of God suffering? This study uses a qualitative method with a doctrinal approach. This research more or less refers to the Encyclical Salvifici Doloris from John Paul II, the Encyclical Spe Salvi from Benedict XVI, and the book Iman Katolik: Buku Informasi dan Referensi from the Indonesian Bishops' Conference. This research found two things, first, Christ's suffering on the cross is proof of God's existence. Second, the God who suffers is strength in the face of suffering.

Solutions to a coupled Cahn–Hilliard-phase-field-crystal model for grain boundary motion

In this paper, we consider an initial–boundary value problem (IBVP) of a coupled Cahn–Hilliard-phase-field-crystal (CH-PFC) system subject to homogeneous Neumann boundary conditions in one spatial dimension. This CH-PFC model couples the composition field of a diffusing species with the crystallographic and can be used to model the diffusion-induced grain boundary migration in crystalline materials. Under suitable assumptions on the coefficients and initial data, we prove that the IBVP possesses a global weak solution. Our existence proof, which contributes to the verification of the model, is only valid in one space dimension.

Connecting descent and peak polynomials

A permutation $\sigma=\sigma_1 \sigma_2 \cdots \sigma_n$ has a descent at $i$ if $\sigma_i>\sigma_{i+1}$. A descent $i$ is called a peak if $i>1$ and $i-1$ is not a descent. The size of the set of all permutations of $n$ with a given descent set is a polynomials in $n$, called the descent polynomial. Similarly, the size of the set of all permutations of $n$ with a given peak set, adjusted by a power of $2$ gives a polynomial in $n$, called the peak polynomial. In this work we give a unitary expansion of descent polynomials in terms of peak polynomials. Then we use this expansion to give an interpretation of the coefficients of the peak polynomial in a binomial basis, thus giving a constructive proof of the peak polynomial positivity conjecture.

A family of condorcet domains that are single-peaked on a circle

AbstractFishburn’s alternating scheme domains occupy a special place in the theory of Condorcet domains. Karpov (2023) generalised these domains and made an interesting observation proving that all of them are single-peaked on a circle. However, an important point that all generalised Fishburn domains are maximal Condorcet domain remained unproved. We fill this gap and suggest a new combinatorial interpretation of generalised Fishburn’s domains which provide a constructive proof of single-peakedness of these domains on a circle. We show that classical single-peaked domains and single-dipped domains as well as Fishburn’s alternating scheme domains belong to this family of domains while single-crossing domains do not.

The Problem of Natural Representation of Reasoning in the Lvov-Warsaw School

ABSTRACT The problem of precise characterisation of traditional forms of reasoning applied in mathematics was independently investigated and successfully resolved by Jaśkowski and Gentzen in 1934. However, there are traces of earlier interests in this field exhibited by the members of the Lvov-Warsaw School. We focus on the results obtained by Jaśkowski and Leśniewski. Jaśkowski provided the first formal system of natural deduction in 1926. Leśniewski also demonstrated in some of his papers how to construct proofs in accordance with intuitively correct principles.

A Constructive Proof of Korn’s Scaled Inequalities for Shells

A Constructive Proof of Korn’s Scaled Inequalities for Shells

Эргономическая сфера как научный феномен и его значение в развитии современного социотехноприродного мира

The article discusses a concept based on the assertion of the existence of an ergonomic sphere (Ergonomicsphere) directly affecting the sustainability of sociotechnical systems in the modern socio-techno-natural world. The basis of this concept lays the understanding that ergonomicsphere can act as a nucleus of sustainability contributing to a paradigm shift in anthropotechnogenic development towards a social-natural one. As a proof of the ergonomicsphere existence, the authors carry out a concept analysis of the systems ergonomicity and their elements; analyse the users’ interest in ergonomics and ergodesign in the context of search activity on the Internet. Particular attention is paid to the interdisciplinary nature of research on the ergonomics epistemology and ergonomic design in the socio-technogenic world. The developed ontological model can form the basis of the paradigm of socio-technobiosphere world development as an ergonomic core of sustainability. As a result of the article, conclusions are drawn about the need to strengthen interdisciplinary integration, which contributes to a more effective study of the modern world and transformation processes of the biosphere.

Travelling modulating pulse solutions with small tails for a nonlinear wave equation in periodic media

Travelling modulating pulse solutions consist of a small amplitude pulse-like envelope moving with a constant speed and modulating a harmonic carrier wave. Such solutions can be approximated by solitons of an effective nonlinear Schrödinger equation arising as the envelope equation. We are interested in a rigorous existence proof of such solutions for a nonlinear wave equation with spatially periodic coefficients. Such solutions are quasi-periodic in a reference frame co-moving with the envelope. We use spatial dynamics, invariant manifolds, and near-identity transformations to construct such solutions on large domains in time and space. Although the spectrum of the linearised equations in the spatial dynamics formulation contains infinitely many eigenvalues on the imaginary axis or in the worst case the complete imaginary axis, a small denominator problem is avoided when the solutions are localised on a finite spatial domain with small tails in far fields.

An in-principle super-polynomial quantum advantage for approximating combinatorial optimization problems via computational learning theory.

It is unclear to what extent quantum algorithms can outperform classical algorithms for problems of combinatorial optimization. In this work, by resorting to computational learning theory and cryptographic notions, we give a fully constructive proof that quantum computers feature a super-polynomial advantage over classical computers in approximating combinatorial optimization problems. Specifically, by building on seminal work by Kearns and Valiant, we provide special instances that are hard for classical computers to approximate up to polynomial factors. Simultaneously, we give a quantum algorithm that can efficiently approximate the optimal solution within a polynomial factor. The quantum advantage in this work is ultimately borrowed from Shor's quantum algorithm for factoring. We introduce an explicit and comprehensive end-to-end construction for the advantage bearing instances. For these instances, quantum computers have, in principle, the power to approximate combinatorial optimization solutions beyond the reach of classical efficient algorithms.

Economic distributions, primitive distributions, and demand recovery in monopolistic competition

We link fundamental technological and taste distributions to endogenous economic distributions of prices and firm size (output, profit) generated under monopolistic competition with heterogeneous productivities as per recent Trade and Industrial Organization models. We derive full equivalence properties for monopoly mark-ups, demand shape, marginal revenue, and profits to match distributions of cost, price, output, and profit under monopolistic competition. Demand and one distribution determine the rest. We provide constructive proofs to recover demand and all distributions from just two (e.g., price and cost distributions uncover demand form), and derive restrictions on distribution pairs. Finally, we extend to analyze mark-up distributions.

CFAuditChain: Audit BlockChain Based On Cuckoo Filter

Abstract Log blockchain can be used to ensure the integrity of log data. However, current methods are facing the problems of throughput mismatch and rough audit granularity. Packaging multiple logs to generate integrity proofs improves throughput but reduces audit granularity, and the auditor can only locate the tampering log packet instead of specific records. This paper proposes CFAuditChain, an audit blockchain based on a cuckoo filter, where the proof of existence (PoE) of the single log is calculated while calculating the integrity proof of packet logs. The PoE is saved in the cuckoo filter and stored using blockchain for immutability. Therefore, the auditor can verify the credibility of each log based on the filter when the integrity proof of a log packet does not match. The theoretical analysis and experimental results show that CFAuditChain provides the granularity of audit logs down to the records level at an acceptable cost.

Multiplicity of positive solutions for a class of nonhomogeneous elliptic equations in the hyperbolic space

The paper is concerned with positive solutions to problems of the type \[ -\Delta_{\mathbb{B}^{N}} u - \lambda u = a(x) |u|^{p-1}\;u + f \text{ in }\mathbb{B}^{N}, \quad u \in H^{1}{(\mathbb{B}^{N})}, \] where $\mathbb {B}^N$ denotes the hyperbolic space, $1< p<2^*-1:=\frac {N+2}{N-2}$ , $\;\lambda < \frac {(N-1)^2}{4}$ , and $f \in H^{-1}(\mathbb {B}^{N})$ ( $f \not \equiv 0$ ) is a non-negative functional. The potential $a\in L^\infty (\mathbb {B}^N)$ is assumed to be strictly positive, such that $\lim _{d(x, 0) \rightarrow \infty } a(x) \rightarrow 1,$ where $d(x,\, 0)$ denotes the geodesic distance. First, the existence of three positive solutions is proved under the assumption that $a(x) \leq 1$ . Then the case $a(x) \geq 1$ is considered, and the existence of two positive solutions is proved. In both cases, it is assumed that $\mu ( \{ x : a(x) \neq 1\}) > 0.$ Subsequently, we establish the existence of two positive solutions for $a(x) \equiv 1$ and prove asymptotic estimates for positive solutions using barrier-type arguments. The proofs for existence combine variational arguments, key energy estimates involving hyperbolic bubbles.

Modelling the industrial production of electric and gas utilities through the $$CIR^3$$ model

AbstractThis work aims to extend previous research on how a trifactorial stochastic model, which we call $$CIR^3$$ C I R 3 , can be turned into a forecasting tool for energy time series. In particular, in this work, we intend to predict changes in the industrial production of electric and gas utilities. The model accounts for several stylized facts such as the mean reversion of both the process and its volatility to a short-run mean, non-normality, autocorrelation, cluster volatility and fat tails. In addition to that, we provide two theoretical results which are of particular importance in modelling and simulations. The first is the proof of existence and uniqueness of the solution to the SDEs system that describes the model. The second theoretical result is to convert, by the means of Lamperti transformations, the correlated system into an uncorrelated one. The forecasting performance is tested against an ARIMA-GARCH and a nonlinear regression model (NRM).

Existence and uniqueness of s-curve segments of tensioned elastica satisfying geometric Hermite interpolation conditions

It has been recently proved that every proper restricted elastic spline is a stable nonlinear spline, and this yields a broad existence proof for stable nonlinear splines. When tension is included in the setup, stable nonlinear splines under tension always exist, but they do not always have the property that each piece (connecting one interpolation point to the next) is an s-curve. Being correlated with the fairness of an interpolating curve, this property is desirable and we conjecture that the framework employed successfully with restricted elastic splines will also work well with nonlinear splines under tension. Our purpose is to prove the following foundational result: Given points P1≠P2, in the plane, along with corresponding unit directions d1,d2 that satisfy d1⋅(P2−P1)≥0 and d2⋅(P2−P1)≥0, there exists a unique s-curve segment of Euler–Bernoulli elastica under tension λ>0 that connects P1 to P2 with initial direction d1 and terminal direction d2.

Row–column duality and combinatorial topological strings

Integrality properties of partial sums over irreducible representations, along columns of character tables of finite groups, were recently derived using combinatorial topological string theories (CTST). These CTST were based on Dijkgraaf-Witten theories of flat G-bundles for finite groups G in two dimensions, denoted G-TQFTs. We define analogous combinatorial topological strings related to two dimensional topological field theories (TQFTs) based on fusion coefficients of finite groups. These TQFTs are denoted as R(G)-TQFTs and allow analogous integrality results to be derived for partial row sums of characters over conjugacy classes along fixed rows. This relation between the G-TQFTs and R(G)-TQFTs defines a row-column duality for character tables, which provides a physical framework for exploring the mathematical analogies between rows and columns of character tables. These constructive proofs of integrality are complemented with the proof of similar and complementary results using the more traditional Galois theoretic framework for integrality properties of character tables. The partial row and column sums are used to define generalised partitions of the integer row and column sums, which are of interest in combinatorial representation theory.

Towards Automated Readable Proofs of Ruler and Compass Constructions

Towards Automated Readable Proofs of Ruler and Compass Constructions

Tensor decompositions on simplicial complexes with invariance

Tensors are ubiquitous in mathematics and the sciences, as they allow to store information in a concise way. Decompositions of tensors may give insights into their structure and complexity. In this work, we develop a new framework for decompositions of tensors, taking into account invariance, positivity and a geometric arrangement of their local spaces. We define an invariant decomposition with indices arranged on a simplicial complex which is explicitly invariant under a group action. We give a constructive proof that this decomposition exists for all invariant tensors, after possibly enriching the simplicial complex. We further define several decompositions certifying positivity, and prove similar existence results, as well as inequalities between the corresponding ranks. Our results generalize results from the theory of tensor networks, used in the study of quantum many-body systems, for example.

Familiarity-taxis: A bilateral approach to view-based snapshot navigation

Many insects use view-based navigation, or snapshot matching, to return to familiar locations, or navigate routes. This relies on egocentric memories being matched to current views of the world. Previous Snapshot navigation algorithms have used full panoramic vision for the comparison of memorised images with query images to establish a measure of familiarity, which leads to a recovery of the original heading direction from when the snapshot was taken. Many aspects of insect sensory systems are lateralised with steering being derived from the comparison of left and right signals like a classic Braitenberg vehicle. Here, we investigate whether view-based route navigation can be implemented using bilateral visual familiarity comparisons. We found that the difference in familiarity between estimates from left and right fields of view can be used as a steering signal to recover the original heading direction. This finding extends across many different sizes of field of view and visual resolutions. In insects, steering computations are implemented in a brain region called the Lateral Accessory Lobe, within the Central Complex. In a simple simulation, we show with an SNN model of the LAL an existence proof of how bilateral visual familiarity could drive a search for a visually defined goal.