Budget Imbalance Criteria for Auctions: A Formalized Theorem

We present an original theorem in auction theory: it specifies general conditions under which the sum of the payments of all bidders is necessarily not identically zero, and more generally not constant. Moreover, it explicitly supplies a construction for a finite minimal set of possible bids on which such a sum is not constant. In particular, this theorem applies to the important case of a second-price Vickrey auction, where it reduces to a basic result of which a novel proof is given. To enhance the confidence in this new theorem, it has been formalized in Isabelle/HOL: the main results and definitions of the formal proof are re- produced here in common mathematical language, and are accompanied by an informal discussion about the underlying ideas.Comment: 6th Podlasie Conference on Mathematics 2014, 11 page

A simplified framework for first-order languages and its formalization in Mizar

A strictly formal, set-theoretical treatment of classical first-order logic is given. Since this is done with the goal of a concrete Mizar formalization of basic results (Lindenbaum lemma; Henkin, satisfiability, completeness and Lowenheim-Skolem theorems) in mind, it turns into a systematic pursue of simplification: we give up the notions of free occurrence, of derivation tree, and study what inference rules are strictly needed to prove the mentioned results. Afterwards, we discuss details of the actual Mizar implementation, and give general techniques developed therein.Comment: Ph.D. thesis, defended on January, 20th, 201

Proving soundness of combinatorial Vickrey auctions and generating verified executable code

Using mechanised reasoning we prove that combinatorial Vickrey auctions are soundly specified in that they associate a unique outcome (allocation and transfers) to any valid input (bids). Having done so, we auto-generate verified executable code from the formally defined auction. This removes a source of error in implementing the auction design. We intend to use formal methods to verify new auction designs. Here, our contribution is to introduce and demonstrate the use of formal methods for auction verification in the familiar setting of a well-known auction

Representation Theorems Obtained by Miningacross Web Sources for Hints

A representation theorem relates different mathematical structures by providing an isomorphism between them: that is, a one-to-one correspondence preserving their original properties. Establishing that the two structures substantially behave in the same way, representation theorems typically provide insight and generate powerful techniques to study the involved structures, by cross-fertilising between the methodologies existing for each of the respective branches of mathematics. When the related structures have no obvious a priori connection, however, such results can be, by their own nature, elusive. Here, we show how data-mining across distinct web sources (including the Online Encyclopedia of Integer Sequences, OEIS), was crucial in the discovery of two original representation theorems relating event structures (mathematical structures commonly used to represent concurrent discrete systems) to families of sets (endowed with elementary disjointness and subset relations) and to full graphs, respectively. The latter originally emerged in the apparently unrelated field of bioinformatics. As expected, our representation theorems are powerful, allowing to capitalise on existing theorems about full graphs to immediately conclude new facts about event structures. Our contribution is twofold: on one hand, we illustrate our novel method to mine the web, resulting in thousands of candidate connections between distinct mathematical realms; on the other hand, we explore one of these connections to obtain our new representation theorems. We hope this paper can encourage people with relevant expertise to scrutinize these candidate connections. We anticipate that, building on the ideas presented here, further connections can be unearthed, by refining the mining techniques and by extending the mined repositories.Comment: In press at IEEEXplor

Pseudo-Canonical Formulae are Classical

An original result about Hilbert Positive Propositional Calculus introduced in [11] is proven. That is, it is shown that the pseudo-canonical formulae of that calculus (and hence also the canonical ones, see [17]) are a subset of the classical tautologies.My work has been partially supported by EPSRC grant EP/J007498/1 and an LMS Computer Science Small GrantCaminati Marco B. - School of Computer Science University of Birmingham Birmingham, B15 2TT United KingdomKorniłowicz Artur - Institute of Informatics University of Białystok Sosnowa 64, 15-887 Białystok PolandGrzegorz Bancerek. Cardinal numbers. Formalized Mathematics, 1(2):377–382, 1990.Grzegorz Bancerek. The ordinal numbers. Formalized Mathematics, 1(1):91–96, 1990.Czesław Byliński. Functions and their basic properties. Formalized Mathematics, 1(1): 55–65, 1990.Czesław Byliński. Functions from a set to a set. Formalized Mathematics, 1(1):153–164, 1990.Czesław Byliński. Basic functions and operations on functions. Formalized Mathematics, 1(1):245–254, 1990.Czesław Byliński. The modification of a function by a function and the iteration of the composition of a function. Formalized Mathematics, 1(3):521–527, 1990.Czesław Byliński. Partial functions. Formalized Mathematics, 1(2):357–367, 1990.Czesław Byliński. Some basic properties of sets. Formalized Mathematics, 1(1):47–53, 1990.Marco B. Caminati. Preliminaries to classical first order model theory. Formalized Mathematics, 19(3):155–167, 2011. doi:10.2478/v10037-011-0025-2.Agata Darmochwał. Finite sets. Formalized Mathematics, 1(1):165–167, 1990.Adam Grabowski. Hilbert positive propositional calculus. Formalized Mathematics, 8(1): 69–72, 1999.Beata Padlewska. Families of sets. Formalized Mathematics, 1(1):147–152, 1990.Piotr Rudnicki and Andrzej Trybulec. Abian’s fixed point theorem. Formalized Mathematics, 6(3):335–338, 1997.Andrzej Trybulec. Domains and their Cartesian products. Formalized Mathematics, 1(1): 115–122, 1990.Andrzej Trybulec. Binary operations applied to functions. Formalized Mathematics, 1 (2):329–334, 1990.Andrzej Trybulec. Defining by structural induction in the positive propositional language. Formalized Mathematics, 8(1):133–137, 1999.Andrzej Trybulec. The canonical formulae. Formalized Mathematics, 9(3):441–447, 2001.Andrzej Trybulec. Classes of independent partitions. Formalized Mathematics, 9(3): 623–625, 2001.Zinaida Trybulec. Properties of subsets. Formalized Mathematics, 1(1):67–71, 1990.Edmund Woronowicz. Relations and their basic properties. Formalized Mathematics, 1 (1):73–83, 1990.Edmund Woronowicz. Relations defined on sets. Formalized Mathematics, 1(1):181–186, 1990.Edmund Woronowicz and Anna Zalewska. Properties of binary relations. Formalized Mathematics, 1(1):85–89, 1990

COVID-19, asthma, and biological therapies: What we need to know

Managing patients with severe asthma during the coronavirus pandemic and COVID-19 is a challenge. Authorities and physicians are still learning how COVID-19 affects people with underlying diseases, and severe asthma is not an exception. Unless relevant data emerge that change our understanding of the relative safety of medications indicated in patients with asthma during this pandemic, clinicians must follow the recommendations of current evidence-based guidelines for preventing loss of control and exacerbations. Also, with the absence of data that would indicate any potential harm, current advice is to continue the administration of biological therapies during the COVID-19 pandemic in patients with asthma for whom such therapies are clearly indicated and have been effective. For patients with severe asthma infected by SARS-CoV- 2, the decision to maintain or postpone biological therapy until the patient recovers should be a case-by-case based decision supported by a multidisciplinary team. A registry of cases of COVID- 19 in patients with severe asthma, including those treated with biologics, will help to address a clinical challenge in which we have more questions than answers

Allergy and coronavirus disease (COVID-19) international survey: Real-life data from the allergy community during the pandemic

Background: The COVID-19 outbreak brought an unprecedented challenge to the world. Knowledge in the field has been increasing exponentially and the main allergy societies have produced guidance documents for better management of allergic patients during this period. However, few publications so far have provided real-life data from the allergy community concerning allergy practice during the COVID-19 outbreak. Therefore, we proposed an international survey on the management of allergic patients during the current pandemic. Methods: We performed an online survey undertaken to reach out the worldwide allergy community by e-mail and social media. The web-based questionnaire contained 24 questions covering demographic data from the participants, clinical practice during this period, and questions related to the new international classification and coding tools addressed for COVID-19. It was circulated for 8 weeks and had anonymous and volunteer context. Results: Data are presented for 635 participants from 78 countries of all continents. Allergists with long-term professional experience were the main audience. As expected, we received many responses as “I have no data” or “I don’t know” to the questions of the survey. However, most with more experience on managing allergic patients during the pandemic agreed that patients suffering from allergic or hypersensitivity conditions have no increased risk of contracting COVID19 or developing SARS CoV-2. Also, participants mentioned that none of the allergy treatments (inhaled corticosteroids, allergen immunotherapy, biological agents) increased the risk of contracting COVID-19 infection including severe presentations. Conclusion: The data presented are a starting point in the process of getting feedback on all the recommendations provided by the allergy societies; it could also be the basis of new strategies to support health professionals while new COVID-19 specific treatments and vaccines are being explored. The information here presented intends to be helpful to the community but represents a course of action in a highly specific situation due to the state of emergency, and it should be helpful to health systems

SANI-Severe Asthma Network in Italy: a way forward to monitor severe asthma

Even if severe asthma (SA) accounts for 5-10% of all cases of the disease, it is currently a crucial unmet need, owing its difficult clinical management and its high social costs. For this reason several networks, focused on SA have been organized in some countries, in order to select these patients, to recognize their clinical features, to evaluate their adherence, to classify their biological/clinical phenotypes, to identify their eligibility to the new biologic therapies and to quantify the costs of the disease. Aim of the present paper is to describe the ongoing Italian Severe Asthma Network (SANI). Up today 49 centres have been selected, widespread on the national territory. Sharing the same diagnostic protocol, data regarding patients with SA will be collected and processed in a web platform. After their recruitment, SA patients will be followed in the long term in order to investigate the natural history of the disease. Besides clinical data, the cost/benefit evaluation of the new biologics will be verified as well as the search of peculiar biomarker(s) of the disease

A contribution to an Auction Theory Toolbox through code and discussion

I am contributing a further mechanization of Vickrey’s theorem on weak equilibrium in second price auctions. Submission 6 of stage 1 is an invitation to discuss on early developments in formalizing a toolbox for auction theory, and I wish to add a further viewpoint. My formalization is in Mizar, which is nearer than other systems to the common mathematical language. Auctions being not yet present in the library, there is the chance of taking the first design decisions: one should put effort into making them suitable to build a large future library on them, and at the same time try his best to reduce as much material as possible to the most common mathematical objects, which are well supported in the existing library. I will illustrate how I faced this task. 1 What has been formalized This AISB submission provides the Mizar formalization of a theorem by Vickrey about weak equilibrium in second price auctions. The mathematics is simply and clearly exposed in [3]. A quick summary follows. The initial data is a vector b, containing the bids of each participant. Given this vector, the dynamics of the auction is simply modeled by assuming that each participant has, in his mind, a precise valuation of the auctioned good, which may or may not coincide with the amount of money he bids. The theorem in question says that, in this regard, the best strategy is to make them coincide, in the ‘weak’ sense: given a random b, changing the bid of a participant to his valuation, the payoff of that participant does not decrease. 1.1 Defining the payoff to express the theorem The best possible payoff for a single participant would be given by winning the whole auctioned good without paying anything, in which case it can be quantified by the subjective valuation v he deems the good worth. Given that, generally, any participant gets a fraction ranging from 0 (for losers) to 1 of the auctioned good, and that, of course, any decent auction scheme will impose at least to the winner(s) to disburse an amount of money, such payoff is defined by vx − p; here v is the valuation, x the fraction of the good obtained, and p the amount paid. x and p depend on all the participants’ bid, and as such each of them is a component of two distinct vectors having the same length as b; x and p are respectively termed the allocations and the payments, and are calculated from b according to the auction algorithm. Thus, the theorem’s wordy statement above can be put into this inequality: v · X II b (i)− P b (i) ≤ v · X II bi (i)− P II bi (i) , (1

Yet another proof of Goedel's completeness theorem for first-order classical logic

A Henkin-style proof of completeness of first-order classical logic is given with respect to a very small set (notably missing cut rule) of Genzten deduction rules for intuitionistic sequents. Insisting on sparing on derivation rules, satisfiability theorem is seen to need weaker assumptions than completeness theorem, the missing request being exactly the rule ~ p --> p, which gives a hint of intuitionism's motivations from a classical point of view. A bare treatment of standard, basic first-order syntax somehow more algebraic-flavoured than usual is also given