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This article provides mathematical proof of the existence of stationary solutions for the coagulation equation including source and efflux terms. We demonstrate the convergence of time dependent solutions to these stationary solutions and highlight the exponential rate of convergence. These properties are analyzed for affine linear coagulation kernels, non-negative source terms and positive efflux rates. Numerical examples are included to demonstrate the predicted convergence behaviour.

Formal theoretical proofs show modeling of stock returns on `everywhere differentiable' continuums always is inappropriate to modeling of rational expectations equilibriums (REE). Simultaneously, modeling of stock returns in discrete time always is robust to modeling of each of REE, or feasibility of deviations from REE. In stated respect, modeling in discrete time results in dichotomous sufficiency conditions for each of conformance with, or deviation of investors' priors from REE. Bane of modeling in continuous time is non-dichotomization of connectedness property of stock prices from evolution of stock prices, a contradiction to the norm that connectedness properties be independent of specific elements that are located in topological spaces. In aggregate, while modeling in continuous time induces positive relations between risk (volatility) and returns, contrary to rational expectations, it is relatively low realizations for volatility that have higher risk of generation of negative returns.

We review academic research on dual-sourcing (or dual-mode) dynamic stochastic inventory models, which study the replenishment of inventory from two sources (or two transportation modes) in the presence of demand uncertainty. The classic assumption, also adopted here, is that the two sources (modes) differ by replenishment leadtimes and costs. We cover both discrete and continuous-time models. A distinguish feature of our review from others is that our review is more technically detailed and provides self-contained proofs of several fundamental results in the dual-sourcing literature. We also highlight theory advanced recently by using asymptotic analysis.

The model in which an individual maximizes the ordinal or cardinal total utility subject to her budget constraint has long been the paradigm of individual choice theory in economics. The ordinal total utility theory has the advantage of utility immeasurability, but the disadvantage of being inconsistent with common sense such as the principle of diminishing marginal utility. The cardinal total utility theory is consistent with the aforementioned common sense, but suffers from the problem of utility measurability. This leads to a dilemma when one develops a utility-maximization model: she either needs to abandon common sense, or needs to postulate measurability for utility. This puzzling riddle, as of today, is still unsolved. The purpose of this paper is to present a new theory — the ordinal marginal utility theory — to solve this puzzle, and offers a complete formal axiomatic proof.

The combinatorial liquidity management problem is formally defined. The nature and complexity of the resulting mathematical program are discussed. Casting the ensuing mathematical program into the physical analog of a network flow model offers a natural and elegant approach for problem formulation. This mitigates model misspecification, particularly, as the problem complexity grows. Efficient network algorithms for proofs of maxflow and mincost satisfaction are developed. A variety of examples are chosen to illustrate the presented concepts. The SAS PROC OPTMODEL is introduced as a powerful tool for solving the network flow mapped liquidity problem.

This is an online appendix of "Sup-inf/inf-sup problem on choice of a probability measure by FBSDE approach". Particularly, we provide proofs of Propositions 1-3 and an example of FBSDEs, in which sgn(Z1); sgn(Z2) are determined by solving the FBSDEs explicitly, for Section III-B in the original paper. Moreover, we present another example of the state variable process satisfying the assumptions in Propositions 1 and 2 and a solution of BSDE in Example 2 in Section III-A.

This supplemental appendix provides the following supporting materials. Section SA contains proofs of the theoretical results in Section 4 of the paper on the size of the new conditional specification test. Section SB provides additional theoretical results on the power of the new test. Section SC provides comparison with some power envelopes through simulations. Section SD collects extra details of the empirical application. The full-text version of this paper can be found at: https://ssrn.com/abstract=3609627.

Dying declarations are declarations of a dying individual on wounds which contributed to his or her death or to injuries being inflicted. Statements by the dead well before death in the Indian Proof Act and not the Announcement of death are not allowed. The main justification for approval of the data is that there is no higher proof. A formal or spoken news release with the necessary information of a deceased individual is the phrase "the death announcement." It is a declaration from the individual who died describing the details of his passing. Anyone who is aware and who acknowledges death is inevitable of what he or she claims to lead to death is the trigger or circumstances of death. The declaration of death is based on the "Nemo moriturus praesumitur lie" proverb that means "a guy won't be lying in his deathbed" to his father. It is a derogation from the rule of proof. Proofs are omitted because they are not trustworthy enough. This is rejected as the checks used on the official proof, e.g. oath and cross examination, are not allowed. The structure of the United Nations renders it useless for anyone who may provide this knowledge not to tell what they believe, but to be tested in a manner that tests the facts. It is an anomaly so long as this testimony does not represent the awful intent of the judicial system and lacks limits because the other complainant to the crime is unwilling, rather than the one who has died afterwards.

As a supplement to Zhu et al. (2020), this document consists of five parts. First, it provides technical proofs of some theoretical results in Zhu et al. (2020). Second, it introduces some preliminary results on stochastic distribution systems with fixed shipment costs. Third, it provides some examples and illustrations on the modified echelon (r,Q) policies (and some related concepts) studied in Zhu et al. (2020). Fourth, it presents some explicit performance bounds that depends on system primitives for the modified echelon (r,Q) policies in stochastic distribution systems. Finally, extensive numerical experiments are conducted to test the effectiveness of the modified echelon (r,Q) policies. Specifically, the numerical studies show that the modified echelon (r,Q) policies perform well with an average gap of about 5% above the cost lower bound, 2) outperform the echelon-stock (r, Q) policy used in Chen and Zheng (1997), and 3) have robust performances with respect to the allocation rule at the warehouse.

The Internet Appendix collects the proofs and additional results that support the main text. The additional theoretical results include a detailed description of special cases and related models and an extension to noisy and misspecified state processes. We also provide an estimator for the number of factors. The additional empirical results consider alternative state processes and discuss the choice of tuning parameters. We also study a portfolio application of our state-varying factors. The extensive simulation section compares the performance relative to alternative latent factor models that allow for time-variation and studies the choice of bandwidth and number of factors with cross-validation arguments. Lastly, we collect the detailed proofs for all the theoretical statements.

The “four-color” theorem seems to be generalizable as follows. The four-letter alphabet is sufficient to encode unambiguously any set of well-orderings including a geographical map or the “map” of any logic and thus that of all logics or the DNA (RNA) plan(s) of any (all) alive being(s). Then the corresponding maximally generalizing conjecture would state: anything in the universe or mind can be encoded unambiguously by four letters. That admits to be formulated as a “four-letter theorem”, and thus one can search for a properly mathematical proof of the statement. It would imply the “four colour theorem”, the proof of which many philosophers and mathematicians believe not to be entirely satisfactory for it is not a “human proof”, but intermediated by computers unavoidably since the necessary calculations exceed the human capabilities fundamentally. It is furthermore rather unsatisfactory because it consists in enumerating and proving all cases one by one. Sometimes, a more general theorem turns out to be much easier for proving including a general “human” method, and the particular and too difficult for proving theorem to be implied as a corollary in certain simple conditions. The same approach will be followed as to the four colour theorem, i.e. to be deduced more or less trivially from the “four-letter theorem” if the latter is proved. References are only classical and thus very well-known papers: their complete bibliographic description is omitted.

In this paper we compare three ways of taking into account the basket skew by using a LVLC (Local Volatility Local Correlation) model. The two first methods were already presented in Langnau (2009) and Guyon and Henry-Labordere (2012). The third method, which is the main result of this paper, reduces the number of used particles by a factor of 10 and was inspired from Muguruza (2019). This article is organized as follows. After a paragraph where we introduce the hypothesis and notations, we briefly present the final objective of the calibration. We then propose 3 solutions which are special cases of the (Guyon, 2017) framework: • Method 1 does not require a conditional expectation calculation (and therefore has no calibration phase) is presented here for comparison • Method 2 is based on the classical calculation of conditional expectations by choosing a Kernel and a bandwidth parameter • Method 3 calculates the conditional expectations required by method 2 using the diffusion implied kernel Then we move on to numerical results paragraph where we present honestly the benefits of the alternative method followed by the appendices where we provide the proofs of the propositions and the details of the calculations allowing to easily implement the diffusion implied kernel.

This is the supplemental material to the paper titled "Common Fund Flows: Flow Hedging and Factor Pricing." It includes many useful and interesting additional empirical results. It also includes the detailed proofs for the theoretical results.

The concept of ‘the core’ originates in cooperative game theory and its introduction to economics in the 1960s as a basis for proofs of existence of general equilibrium is one of the earliest attempts to use game theory to address big questions in economics. Discovery of the core was met with enthusiasm among the community of economic theorists at the time. However, use of the core eventually waned and the concept faded into the backdrop of economic theory. This paper makes use of unpublished correspondence between Herbert Scarf, Lloyd Shapley, and Martin Shubik, as well as other archival and secondary resources, to provide an account of the development of the core and the trajectory of this concept, including those who developed it, after its initial appearance. It is found that the core’s eventual decline is explained by the combined effect of the slowing general equilibrium research program in the 1970s, the increasing prominence of non-cooperative game theory, and subtle issues with the concept that shaped Scarf and Shubik’s research programs after the 1960s.

This study applies Fama-French-style factor loading analysis to cryptocurrency financial performance data to determine the originality of 32 reportedly novel consensus algorithms (“proofs”) and 20 hybrid consensus mechanisms as compared to conventional proof-of-work and proof-of-stake using a sample of 302 cryptocurrencies. Only 14 out of 32 new consensus algorithms and 12 out of 20 hybrid mechanisms are found to be truly original. Innovative consensus protocols are not associated with superior returns while original hybrid solutions are. The findings allow investors to select coins with original “proofs” and to explore performance implications of consensus algorithms. For future research, the applicability of market, size, proof and age factors for risk and attribution analysis of cryptocurrency markets is evidenced.

This supplementary Technical Appendix contains formal proofs of the propositions which are stated in Anyfantaki, S., S. Arvanitis, S., Th. Post, Th. and N. Topaloglou, 2019, 'Stochastic Bounds for Portfolio Analysis', available at SSRN:https://ssrn.com/abstract=3181869 or http://dx.doi.org/10.2139/ssrn.3181869, together with additional results and references.

Full paper is available at: https://ssrn.com/abstract=3233422 This online appendix contains all the proofs of the main results in Han and Lee (2018), technical assumptions, and additional simulation results. Section A contains the proofs of the identification/non-identification results. Section B collects the proofs in the asymptotic theory of the sieve ML estimators, related technical assumptions, and other expressions. Finally, Section C shows more simulation results.

There exist several modeling languages and frameworks for formal representation and analysis of systems. This paper aims to present an in-depth survey on Event-B based model checking. Event B is an evolution from the earlier B language. It has drawn reasonable attention both for industrial and academic communities due to its openness towards the iterative refinements which validate designs starting from a high-level abstraction of the system. The Rodin is a tool that provides an open-source implementation platform for Event-B based model checking and mathematical proofs. Researchers across the globe are using Event B for industry automation, design of embedded and safety-critical systems, and in many more areas. The manuscript presented a well-researched classification considering diverse broad fields where Event-B modeling is applied. Accordingly, research gaps have been identified to highlight future research directions and application domains for event B.

This supplemental material provides the detailed arguments of some omitted proofs in the example analyzed in Marins-da-Rocha, Phan and Vailakis (2019) (henceforth MPV).

SCSA is the abbreviation of secure cloud storage auditing. Cloud storage auditing is a mechanism to ensure the integrity of data stored in cloud. Many Individuals and organizations depend on cloud to store their vast amount of data. After storing their data, client loses the control over their data, So it is natural that client become doubtful about the existence of their data which is stored in the cloud. Thus, this auditing scheme is introduced. This scheme helps the client to check whether their data is altered by any other third party, But this auditing scheme is also vulnerable to attacks. A secret key is required to prove the existence of original data in cloud. But that secret key is not secured properly, so in case if the key is lost, an attacker can make fake proofs of the data and thus data loss incidents can be made hidden from the clients, To avoid that, a solution is presented in this paper that is periodic key update. More efficient algorithm is used for updating key, and also this paper proposes two new algorithm for auditing purposes, In addition duplication of data is also avoided.

This online appendix includes supplemental materials for "Network-motivated Lending Decisions: A Rationale for Forbearance Lending'' by Ogura, Okui, and Saito. Online Appendix 1 contains the proofs of the propositions. Online Appendix 2 presents a numerical example for the theoretical model. Online Appendix 3 derives the bias property of the OLS estimator applied to the spatial autoregressive model. Online Appendix 4 presents the formula for standard errors for the OLS estimator, which are adjusted to address the problem that the demand influence coefficients are estimated regressors. Online Appendix 5 presents additional empirical results to examine the robustness of the empirical results in the main text.

We present an elementary approach to fundamental theorems of asset and derivative pricing for a wide class of jump-diffusion markets in which there are credit risks, funding risks, and collateral. The approach yields new and intuitive proofs of both the first and second Fundamental Theorems of Asset Pricing. We give examples of how the framework incorporates the gamut of industry pricing techniques from simple Black-Scholes pricing to recent results known as Funding Valuation Adjustments (FVA). The main contribution is the insight and intuition into the nature of the fundamental theorems that is afforded by the simplicity of the setup and the consequent reliance on only the most basic results from linear algebra.

This note contains the supplements to Li and Linton (2019). Section A contains extensive simulation studies. Section B presents the methodology and a detailed numerical algorithm to select the tuning parameters. Section C presents additional empirical studies. Section D contains all mathematical proofs.

Contained herein are detailed proofs of the lemmas that support the theorems presented in the main paper under the same title. In addition, this supplemental appendix also contains an additional theorem (and accompanying proof) that is referenced, but not included, in the main paper.

This online appendix is structured as follows. Section S.1 contains some auxiliary lemmas and proofs for Section 4. Section S.2 develop supporting theoretical results for Section 5. Section S.3 reports additional simulation results. Some tables appear at the end.

Pursuing abstractness and generality, a number of books and articles in econometrics rely heavily on standard algebraic proofs. However, most of the theorems proved in such a way have a strong geometric appeal. This paper demonstrates how shorter, less technical and even more beautiful the proofs can be when they are based on geometric theorems. We show that this technique can be extended to explain concepts in probability and statistics. Although most of the theorems and ideas are not completely new and there are even a few works on the geometry in econometrics and statistics, this paper introduces alternative explanations and more general results in some cases. Further, there are algebraic proofs provided parallel to geometric ones and illustrations which are our own work.

תקציר בעברית: חיבור זה סובב סביב פשרה ומהותה של ההוכחה המשפטית הנוגעת בקביעת עובדות . אפתח במספר הערות מקדימות: ראשית, הנושא שלפנינו אינו פשוט, אלא מורכב למדיי, ומצריך עיון בלתי שטחי בתופעת ההוכחה וההסבר בשטחים אינטלקטואליים אחרים. שנית, אודות נושא זה לא נכתב אלא מעט . כאן חשוב להדגיש. במהלך כ-שלושים השנים האחרונים החלה להתפתח ספרות תיאורטית ענפה, אינטרדיספלינארית באופייה, העוסקת במימד האפיסטמולוגי של דיני הראיות, בפרט בשיטה האנגלו-אמריקאית . כך לדוגמא קיים דיון רחב היקף סביב השאלה בדבר מהותה של הסתברות שראויה לשמש, אם בכלל, קובעי עובדות, בפרט שופטים במלאכת הסקת מסקנות עובדתיות . כמו כן קיים ניתוח על אודות קדם ההנחות ההכרתיות של המסורת שבתוכה מתפקדים דיני הראיות . כן מוצעים ומוצגים מודלים אנליטיים ונורמטיביים להבנה ,פירוש או שינוי ועדכון של דיני הראיות הנוהגים . כל זאת תוך שימוש בתובנות מתחומי תורת ההחלטות סטטיסטיקה ותורת המשחקים, סוציולוגיה, פסיכולוגיה קוגניטיבית ,פילוסופיה של ההכרה ותחומים נוספים. כל אלו סובבים סביב העניין שלפנינו אך אינם מציבים אותו כנושא הישיר של בחינתם. עניינו כאן בלוגיקה להבדיל מן האפיסטמולוגיה. עניינו בתכניו המופשטים של מושג התקפות בדיני ההוכחה להבדיל מהצגה של אמצעים ההכרה ורכישת הידע השיפוטיים . נקודת מבטינו מוסבת לעבר ב"צורות" או "האופנים" המופשטים ביותר של המעבר מראיות לעובדות להבדיל מאופן הטיפול בראיות מסוימות או בסוגים מיוחדים של היסקים עובדתיים (לפיכך לא נטפל במישרין באבחנות אפשריות שבין הסקה נסיבתית להסקה על יסוד עדויות ישירות ,חזקות או הודאת בעל הדין או הנאשם)שלישית, ככלל אתמקד כפי שכותרתו של המאמר מרמזת בהשוואת דפוס ההוכחה המשפטית עם זה המתמטי-עיוני וזה הנוהג במדעי הטבע, (ובמדעי ההתנהגות והחברה ככל שהם מציבים את דגם ההוכחה של המדעים המדויקים כמופת של הוכחה ). רק כבדרך אגב אעיר כמה הערות באשר לרעיונות בדבר השוואת ההיסק המשפטי עם דגם ההוכחה וההסבר במדעי הרוח ובשטחים אינטלקטואליים אחרים העוסקים בפרשנות ובהבניה של אפיזודות אנושיות.ורביעית מה שייכתב כאן אינו יותר מהערות תיאורטיות ראשוניות שאינן בוודאי עולות לכדי עמדה שיטתית ומלוטשת באשר לנושא. מכל מקום עיקר חשיבות הדברים, היא לעורר מחשבה ולהפנות את הדעת לסוגיה שטרם זכתה לעיון שלעניות דעתי לו היא ראויה. נפנה אם כך לגוף העניין עצמו In this article I will seek to discuss the essence of legal proof in relation to fact-finding; in particular, a facet thereof that would seem to dovetail with scientific research. I refer, in this regard, to the manner in which the law establishes generalizations (i.e., legal revelations) regarding reality, which it then uses for fact-finding in a given instance; as well as the manner in which the law uses such generalizations for specific factual deduction. The principle intuition that our article seeks to contend with is how the law uses a sort of approximation of the scientific process within the constraints of legal practicality. Essentially, a sort of “layman’s science” that attempts to accomplish the same goals as science - only, not as well, and less methodically. The conclusion we will seek to show in this article is that the manner in which the law derives generalizations about reality, and the manner in which it uses same, fundamentally differs from the manner in which science is conducted. Theories in natural sciences (and largely with respect to social sciences) are framed as generally as possible. Legal proof, on the other hand, is specific . In fact, the most common form of induction used is the cognitive analogy, wherein the discretion employed is far more similar in its characteristics to clinical discretion, such as that used in the field of medicine, then to those typically employed in the fields of natural sciences.

These supplemental materials include extensions and details on the model used in the main text, as well as on its solution. An abbreviated overview of the related literature is also offered. Proofs for the lemmas, propositions, theorems and corollaries stated in the main text are collected here too.

Recently, two types of innovative estimators of the quadratic variation (QV), namely RVB and RVQ estimators, are established in Yu (2020) for an asset's log-price process permitting time-varying spot volatility. Both RVB and RVQ utilize either higher-order- or cross-terms of log-return and hence they have been demonstrated having better convergence property than existing ones (e.g. the widely-known realized volatility). However, the proof of the convergence property of RVB/RVQ is rather cumbersome and algebraically complicated, and the restriction on spot volatility (being a deterministic function of time) in the log-price process is a little bit strong. This article mainly addresses two issues: (1)relax the restriction for asset's price process; (2)substantially shorten the original proofs by alternatively employing two useful lemmas (called high-order It^o isometry and adjoint It^o identity) developed in this study.

This supplement provides proofs of all propositions in [Tomohiko Kawamori and Toshiji Miyakawa, "Equivalent conditions for the existence of an efficient equilibrium in coalitional bargaining with externalities and renegotiations," Operations Research Letters 45(5), 427-430, 2017].

Contained herein are detailed proofs of all the Lemmas that support the main Theorems discussed in the paper, "Simple Estimators for GARCH models." Original paper can be found at: https://ssrn.com/abstract=2897867

We reconsider the theory of Thompson aggregators proposed by Marinacci and Montrucchio. First, we prove a variant of their Recovery Theorem estabilishing the existence of extremal solutions to the Koopmans equation. Our approach applies the constructive Tarski-Kantorovich Fixed Point Theorem rather than the nonconstructive Tarski Theorem employed in their paper. We verify the Koopmans operator has the order continuity property that underlies invoking Tarski-Kantorovich. Then, under more restrictive conditions, we demonstrate there is a unique solution to the Koopmans equation. Our proof is based on υ0- concave operator techniques as first developed by Kransosels'kii. This differs from Marinacci and Montrucchio's proof as well as proofs given by Martins-da-Rocha and Vailakis.

Proofs and Technical Appendix for paper: Spectral characterization of the non-independent increment family of Alpha-Stable processes that generalize Gaussian process models.

This appendix to "Stability, Strategy-Proofness, and Cumulative Offer Mechanisms" (available at https://ssrn.com/abstract=3120463) contains a proof of the sufficiency result (Theorem 4) in Section 3 and all proofs for Section 4.

One of the most debated and controversial topics amongst legal scholars has been the role of probability in the evaluation of legal proofs conducted in the fact finding process which is typical of courts. This matter is especially important in criminal proceedings, where the standard of proof is set to “beyond any reasonable doubt.” In international criminal tribunals, the governing principle is the “free evaluation of evidence” which means that judges are not obliged to respect any kind of rule on how to evaluate evidence and can therefore pick the approach that they consider being the best fit to evaluation. The ad-hoc tribunals and the International Criminal Court so far have not stated a preference for any kind of mathematical or non-mathematical approach to evidence and for this reason an analysis of their case law is inevitably crucial in order to understand which method was applied by the judges in each specific case. This essay will first outline the main differences between mathematical and non-mathematical approaches to evidence. The focal flaws about methods applying mathematical probability theories will be mentioned and it will be argued why this method cannot be applied by the International Criminal Court. Then the non-mathematical approach to evidence will be delineated and specific attention will be given to Cohen’s system of inductive probability. Consequently, the reasoning of the International Criminal Court in a selection of cases will be studied in order to determine which approach was picked by the Court. The author will seek to prove the hypothesis that the International Criminal Court applied a reasoning of induction by elimination as delineated in Cohen’s theory. This Essay will aim at answering the following questions: has the International Criminal Court adopted Cohen’s Alternative Hypothesis Approach in the Al-Bashir and Ngudjolo cases? Is this a suitable approach to evidence in the ICC? What are the problems connected to this kind of approach? Due to the vast academic work on the matter, the author has decided not to describe comprehensively all the existing theories on how to evaluate evidence but to focus on what she thought were the most relevant theories applicable to three ICC cases.

Roughly half of all professionally managed funds globally employ optimized portfolio design principles that are applications of Grinold’s “Fundamental Law of Active Management.” These include: Invest in many securities, use many factors to forecast return, trade frequently, and optimize with minimal constraints. We show with simple examples followed by rigorous simulation proofs that these proposals are invalid and self-defeating. This is because estimation error and required economically valid constraints are ignored in derivations. These flawed principles have been unchallenged by academics and practitioners for nearly twenty-five years and may adversely impact a trillion dollars or more in current fund management.

In this paper we present some new results for the dynamic agent model by Iossa and Rey (2014, "Building Reputation for Contract Renewal: Implications for Performance Dynamics and Contract Duration,'' Journal of the European Economic Association, 12, 549−574) while also correcting some errors in that article. Iossa and Rey study the performance of an agent who repeatedly receives multi-period contracts and determine the optimal duration of such contracts in the context of an infinitely repeated multi-period agent model. We amend the characterization of the unique Markov perfect equilibrium for this model. In addition, we review the original welfare analysis of the model and either provide corrected proofs when possible or provide counterexamples. Our counterexamples overturn the main comparative statics results of the original analysis. We demonstrate that both the agent's optimal investment decision and the optimal contract duration depend non-monotonically on the information persistence and the agent's discount factor. In the final part of the analysis, we establish new results on the agent's optimal investment decision.

We carried out an empirical investigation among producers of packaging machines collecting information about their rivals, i.e., those few competitors which they closely monitor. We found interesting regularities that take geographically clustered firms apart from isolated firms, but we could not explain our empirical findings by means of existing theories. By constructing an agent-based model we were able to formulate a simple and plausible mechanism for rival selection which, once combined with well-known aspects of knowledge representation among clustered and isolated firms, is able to generate the empirically observed facts. We submit that this case is exemplary in showing what agent-based models can do, namely, providing sufficiency proofs that help theory-building.

Linear regression models form the cornerstone of applied research in economics and other scientific disciplines. When conditional heteroskedasticity is present, or at least suspected, the practice of reweighting the data has long been abandoned in favor of estimating model parameters by ordinary least squares (OLS), in conjunction with using heteroskedasticity consistent (HC) standard errors. However, we argue for reintroducing the practice of reweighting the data, since doing so can lead to large efficiency gains of the resulting weighted least squares (WLS) estimator over OLS even when the model for reweighting the data is misspecified. Efficiency gains manifest in a first-order asymptotic sense and thus should be considered in current empirical practice. Crucially, we also derive how asymptotically valid inference based on the WLS estimator can be obtained even when the model for reweighting the data is misspecified. The idea is that, just like the OLS estimator, the WLS estimator can also be accompanied by HC standard errors without knowledge of the functional form of conditional heteroskedasticity. A Monte Carly study demonstrates attractive finite-sample properties of our proposals compared to the status quo, both in terms of estimation and making inference.

The average worker in the US needs to work more than a year to earn his or her CEO’s daily wage. The well-accepted justification among economists for these huge wage gaps is the necessity to achieve economic efficiency, in terms of efficient allocation in the labor market and incentivizing employees to put forth maximum effort. Building on the idea that the level of effort any individual can invest is bounded, we provide a compact mathematical proof that challenges the well-accepted economic notion that extremely high gaps in net wages and wealth distribution are the lesser evil because they ensure economic efficiency. We conclude suggesting an economically efficient solution that does not limit gross wages, but still reduces socioeconomic inequalities.

There are various schools of Mathematical philosophy. However, none of them can founded our Mathematics subject by itself only. At the same time, there are two types of mathematical proof styles: Dialectic and algorithm Mathematical proof. The relation between proof and philosophy is to study philosophical problems through building up mathematics model. Therefore proof is important to Hong Kong Secondary education. However, teachers should explain the connectivity between mathematics subjects and others like physics so that lessons will become more interesting rather than a technical one.

In an earlier paper, the researcher explained how mathematics is not only a technical subject but also a cultural one. As such, mathematical proofs and definitions, instead of simply numerical calculations, are essential for students when learning the subject. Hence, there must be a change in the pedagogies of Hong Kong’s local teachers. The author suggests three alternative ways to teach mathematical philosophy through infinity. These alternatives are as follows: 1. teach the concept of a limiting value in formalism through storytelling, 2. use geometry to intuitively learn infinity through constructivism, and 3. implement schematic stages for proof by contradiction. Simultaneously, teachers should also be aware of the difficulties among students in understanding different abstract concepts. These challenges include the following: 1. struggles with the concept of a limit, 2. mistakes in computing infinity intuitively, and 3. challenges in handling the method of proof by contradiction. By adopting these alternative approaches, teachers can provide the necessary support to pupils trying to comprehend the previously mentioned difficult mathematical ideas and ultimately transform students’ beliefs (Rolka et al., 2007). One can analyse these changed beliefs against the background of conceptual change. According to Davis (2001) & King,A, p.30 ‘this change implies conceiving of teaching as facilitating, rather than managing learning and changing roles from the sage on the stage to a guide on the side’. As a result, Hong Kong’s academic results in mathematics should hopefully improve.

Contrary to the common opinion that Kierkegaard is a fideist, there are in fact good proofs that he offers arguments (of an ethical and existential nature) in favor of the belief in God’s existence. However, this type of philosophical justification is not classical. Evaluated through the categories of modern epistemology it may be called rather externalist than internalist. The advantage of this approach to the problem of the arguments for God’s existence is its egalitarian, eliberating and formative consequences. In addition to that, a result of this view on the belief in God (and the knowledge of his attributes) is the fact that they do not seem to depend in any way by the contingent developments of science or society; therefore, they cannot be affected and eventually deteriorated by these factors.

We provide an alternative proof of the strong duality theorem whose main step is a proposition which says that every canonical linear programming minimization problem whose image under its objective function of the set of feasible solutions is non-empty and bounded below has an optimal solution. Unlike earlier proofs, this proof neither uses the simplex method, nor does it use Farkas’s lemma. We also use this proposition to obtain an independent proof of the Farkas’s lemma.

In fall 1935, Abraham Wald presented an existence proof for a general equilibrium of exchange model to Karl Menger’s Mathematical Colloquium in Vienna. Due to limited space, the paper could not be printed in the eighth proceedings of the Colloquium (the Ergebnisse) published in spring 1937 but was scheduled for the ninth issue of the series. After the annexation of Austria to Nazi Germany in March 1938 however, Menger’s Colloquium ended and the proof never appeared in print. Nor did Wald, after he fled to the U.S. and launched a career in statistics, pursue the diffusion of his proof. Since the proof never appeared, there has been continued speculation about whether Wald’s missing proof employed a fixed-point theorem. To date this question has remained unanswered. After his sudden death in 1950, only Wald’s preliminary proof of 1934 was translated into English for Econometrica. When thus Arrow, Debreu, and McKenzie in 1954 referred in their own fixed-point proofs only to Wald’s preliminary published version, his 1935 proof was forgotten. This did not change when economists and historians of economics, the authors included, reconstructed Wald’s contribution. The authors’ new evidence, however, establishes that the missing proof employed a fixed-point theorem. This article tells the story of Wald’s lost equilibrium proof.

The Weibull distribution is widely used in natural sciences and engineering, especially in material reliability. For one is often confronted to censored data sample in these fields, the truncated Weibull distribution functions and moments can be of great interest. McEwen and Parresol (1991) became the main reference over the years as they exhibit the moments expressions of the truncated distribution for both the two-parameter and three-parameter Weibull random variables. Unfortunately, a mistake has been made when writing the doubly-truncated three-parameter Weibull probability distribution function and it naturally led to wrong results. In this article, we aim at correcting this error and giving explicit mathematical demonstrations of how moments of truncated Weibull distributions can be computed. In order to be as clear as possible we give the explicit expressions and proofs for the two-parameter Weibull one-sided and doubly-truncated moments as well as for the three-parameter Weibull distribution, although every result can be derived from the doubly-truncated three-parameter Weibull distribution moments. Besides, an R code with a function computing these moments is given in the last section.

An open distributed system can be secured by requiring participants to present proof of work and rewarding them for participation. The Bitcoin digital currency introduced this mechanism, which is adopted by almost all contemporary digital currencies and related services.A natural process leads participants of such systems to form pools, where members aggregate their power and share the rewards. Experience with Bitcoin shows that the largest pools are often open, allowing anyone to join. It has long been known that a member can sabotage an open pool by seemingly joining it but never sharing its proofs of work. The pool shares its revenue with the attacker, and so each of its participants earns less.We define and analyze a game where pools use some of their participants to infiltrate other pools and perform such an attack. With any number of pools, no-pool-attacks is not a Nash equilibrium. We study the special cases where either two pools or any number of identical pools play the game and the rest of the participants are uninvolved. In both of these cases there exists an equilibrium that constitutes a “tragedy of the commons” where the participating pools attack one another and earn less than they would have if none had attacked.For two pools, the decision whether or not to attack is the miner’s dilemma, an instance of the iterative prisoner’s dilemma. The game is played daily by the active Bitcoin pools, which apparently choose not to attack. If this balance breaks, the revenue of open pools might diminish, making them unattractive to participants.

The Appendix provides proofs for the major results in the paper ''Flexible System Design: A Perspective from Service Levels''.