Trivial Proof Research Articles

Bilevel linear optimization belongs to NP and admits polynomial-size KKT-based reformulations

It is a well-known result that bilevel linear optimization is NP-hard. In many publications, reformulations as mixed-integer linear optimization problems are proposed, which suggests that the decision version of the problem belongs to NP. However, to the best of our knowledge, a rigorous proof of membership in NP has never been published, so we close this gap by reporting a simple but not entirely trivial proof. A related question is whether a large enough “big M” for the classical KKT-based reformulation can be computed efficiently, which we answer in the affirmative. In particular, our big M has polynomial encoding length in the original problem data.

On the pillars of Functional Analysis

Many authors consider that the main pillars of Functional Analysis are the Hahn–Banach Theorem, the Uniform Boundedness Principle and the Open Mapping Principle. The first one is derived from Zorn’s Lemma, while the latter two usually are obtained from Baire’s Category Theorem. In this paper we show that these three pillars should be either just two or at least eight, since the Uniform Boundedness Principle, the Open Mapping Principle and another five theorems are equivalent, as we show in a very elemental way. Since one can give an almost trivial proof of the Uniform Boundedness Principle that does not require the Baire’s theorem, we conclude that this is also the case for the other equivalent theorems that, in this way, are simultaneously proved in a simple, brief and concise way that sheds light on their nature.

Gaussian noise sensitivity and Fourier tails

We observe a subadditivity property for the noise sensitivity of subsets of Gaussian space. For subsets of volume 1/2, this leads to an almost trivial proof of Borell’s Isoperimetric Inequality for ρ = cos( π/2l), l ∈ N. Rotational sensitivity also easily gives the Gaussian Isoperimetric Inequality for volume-1/2 sets and a.8787-factor UG-hardness for Max-Cut (within 10−4 of the optimum). As another corollary we show the Hermite tail bound $$||{f^{ > k}}||_2^2 \geqslant \Omega (Var[f]).\frac{1}{{\sqrt k }}for:{R^n} \to \{ - 1,1\} $$ . Combining this with the Invariance Principle shows the same Fourier tail bound for any Boolean f: {−1, 1} n → {−1, 1} with all its noisy-influences small, or more strongly, that a Boolean function with tail weight smaller than this bound must be close to a junta. This improves on a result of Bourgain, where the bound on the tail weight was only $$\frac{1}{{{k^{1/2 + o(1)}}}}$$ . We also show a simplification of Bourgain’s proof that does not use Invariance and obtains the bound $$\frac{1}{{\sqrt k {{\log }^{1.5}}k}}$$ .

SURREAL TIME AND ULTRATASKS

AbstractThis paper suggests that time could have a much richer mathematical structure than that of the real numbers. Clark & Read (1984) argue that a hypertask (uncountably many tasks done in a finite length of time) cannot be performed. Assuming that time takes values in the real numbers, we give a trivial proof of this. If we instead take the surreal numbers as a model of time, then not only are hypertasks possible but so is an ultratask (a sequence which includes one task done for each ordinal number—thus a proper class of them). We argue that the surreal numbers are in some respects a better model of the temporal continuum than the real numbers as defined in mainstream mathematics, and that surreal time and hypertasks are mathematically possible.

Linear spin-2 fields in most general backgrounds

We derive the full perturbative equations of motion for the most general background solutions in ghost-free bimetric theory in its metric formulation. Clever field redefinitions at the level of fluctuations enable us to circumvent the problem of varying a square-root matrix appearing in the theory. This greatly simplifies the expressions for the linear variation of the bimetric interaction terms. We show that these field redefinitions exist and are uniquely invertible if and only if the variation of the square-root matrix itself has a unique solution, which is a requirement for the linearized theory to be well defined. As an application of our results we examine the constraint structure of ghost-free bimetric theory at the level of linear equations of motion for the first time. We identify a scalar combination of equations which is responsible for the absence of the Boulware-Deser ghost mode in the theory. The bimetric scalar constraint is in general not manifestly covariant in its nature. However, in the massive gravity limit the constraint assumes a covariant form when one of the interaction parameters is set to zero. For that case our analysis provides an alternative and almost trivial proof of the absence of the Boulware-Deser ghost. Our findings generalize previous results in the metric formulation of massive gravity and also agree with studies of its vielbein version.

POLIFORMNOST KAO NASTAVNI, NAUČNI I FILOZOFSKI PRINCIP

<p>U ovom spisu, autori su razmotrili dijaloškim putem veoma aktualno pitanje misaonog susreta filozofije i matematike. U svom kolegijalnom diskursu, oni su pošli od metodološke pozicije, da je poliformnost nezaobilazno načelo, koje se bazira na dijalektičkom zakonu negacije, koji je u nauci filozofije i matematici prisutan od Aristotelovog vremena do danas. Ključno esencijalno pitanje ovog naučnog i filozofskog fenomena je u njegovom permanentnom insistiranju na integrirnom sagledavanju raznovrsnih pristupa hermeneutičkog razumijevanja i rezoniranja pojmova, posebno u proučavanju naučnih, nastavnih ili filozofskih fenomena. Proučavajući i istraživajući mogućnosti primjenjivanja ovog načela, autori su došli do slijedećeg rezultata: Ako je nesporno da je očiglednost (prezentovanje trivijalnim dokazivanjem primjenom logičkih zakona), princip, a <strong>permanencije </strong>(očuvanje formalnih zakonitosti) utemeljene na očiglednim dokazivanjima, također princip, onda je svaki lanac konjukcija konačnog broja, očiglednih naučnih, metodičkih, filozofskih dokazivanja, ili višestruka konjukcija takvih dokazivanja naučno metodičko ili filozofsko načelo.</p>

A Multinomial Theorem for Hermite Polynomials and Financial Applications

Different aspects of mathematical finance benefit from the use Hermite polynomials, and this is particularly the case where risk drivers have a Gaussian distribution. They support quick analytical methods which are computationally less cumbersome than a full-fledged Monte Carlo framework, both for pricing and risk management purposes. In this paper, we review key properties of Hermite polynomials before moving on to a multinomial expansion formula for Hermite polynomials, which is proved using basic methods and corrects a formulation that appeared before in the financial literature. We then use it to give a trivial proof of the Mehler formula. Finally, we apply it to no arbitrage pricing in a multi-factor model and determine the empirical futures price law of any linear combination of the underlying factors.

The chaos game revisited: Yet another, but a trivial proof of the algorithm’s correctness

The aim of this work is to explain why the most popular algorithm for approximating IFS fractals, the chaos game, works. Although there are a few proofs of the algorithm’s correctness in the relevant literature, the majority of them utilize notions and theorems of measure and ergodic theories. As a result, paradoxically, although the rules of the chaos game are very simple, the logic underlying the algorithm seems to be hard to comprehend for non-mathematicians. In contrast, the proof presented in this work uses only fundamentals of probability and can be understood by anyone interested in fractals.

On lines, joints, and incidences in three dimensions

We extend (and somewhat simplify) the algebraic proof technique of Guth and Katz (2010) [9], to obtain several sharp bounds on the number of incidences between lines and points in three dimensions. Specifically, we show: (i) The maximum possible number of incidences between n lines in R 3 and m of their joints (points incident to at least three non-coplanar lines) is Θ ( m 1 / 3 n ) for m ⩾ n , and Θ ( m 2 / 3 n 2 / 3 + m + n ) for m ⩽ n . (ii) In particular, the number of such incidences cannot exceed O ( n 3 / 2 ) . (iii) The bound in (i) also holds for incidences between n lines and m arbitrary points (not necessarily joints), provided that no plane contains more than O ( n ) points and each point is incident to at least three lines. As a preliminary step, we give a simpler proof of (an extension of) the bound O ( n 3 / 2 ) , established by Guth and Katz, on the number of joints in a set of n lines in R 3 . We also present some further extensions of these bounds, and give a trivial proof of Bourgain's conjecture on incidences between points and lines in 3-space, which is an immediate consequence of our incidence bounds, and which constitutes a much simpler alternative to the proof of Guth and Katz (2010) [9].

New axioms for rigorous Bayesian probability

By basing Bayesian probability theory on flve axioms, we can give a trivial proof of Cox's Theorem on the product rule and sum rule for conditional plausibility without assuming continuity or difierentiablity of plausibility. Instead, we extend the notion of plausibility to apply to unknowns, giving them plausi- ble values. Thus, we combine the best aspects of two approaches to Bayesian probability theory, namely the Cox-Jaynes theory and the de Finetti theory. The problem of developing the foundations of Laplacian or Bayesian probability has led to many difierent approaches, which are mainly variations on two. The flrst, which is our main concern is due to R. T. Cox (1961). The second is due to Bruno de Finetti (1974), de Finetti (1975) and a concise general treatment is in Regazzini (1985). We flnd that by combining aspects of both, we can substantially weaken the assumptions and arrive at a proof of the problematic Cox theorem which is very simple, in fact trivial, as it should be. This triviality is of the utmost importance. The foundational axioms assumed by de Finetti for conditional expectation (his conditional prevision) require a positive normalized linear functional on a partially ordered commutative algebra (with identity element and over the reals) of objects more general than random variables which he calls \random quantities and which we will simply call \unknowns or \unknown numerical quantities. Even though it is not explicitly pointed out axiomatically, de Finetti, in his discussion in the flrst two chapters of his book, assumes that the ordering of the lattice of idempotents is inherited from that of the algebra. To deflne his conditional prevision, he requires a coherence condition (see also Regazzini (1985)) equivalent to a quadratic minimization property assumed to be satisfled by the choice of previsions. We merely assume the existence of some form of rules and demonstrate that if the assignment of plausible value (what de Finetti calls \prevision) is required to be logically consistent, then the actual rules themselves are uniquely determined. In particular, as a conse- quence, the rules of conditional expectation in Bayesian probability theory are uniquely determined by the requirement of logical consistency. Thus we flnd a trivial proof of the Cox Theorem, which properly understood, is a uniqueness theorem, and we show that the theory of de Finetti is a consequence. That is, de Finetti assumes his coherence condition, meaning existence of a positive linear functional which also satisfles a cer- tain quadratic minimization criteria, and he uses the quadratic minimization criteria to

The Number of 0-1-2 Increasing Trees as Two Different Evaluations of the Tutte Polynomial of a Complete Graph

If $T_{n}(x,y)$ is the Tutte polynomial of the complete graph $K_n$, we have the equality $T_{n+1}(1,0)=T_{n}(2,0)$. This has an almost trivial proof with the right combinatorial interpretation of $T_{n}(1,0)$ and $T_{n}(2,0)$. We present an algebraic proof of a result with the same flavour as the latter: $T_{n+2}(1,-1)=T_n(2,-1)$, where $T_{n}(1,-1)$ has the combinatorial interpretation of being the number of 0–1–2 increasing trees on $n$ vertices.

Criteria for the density property of complex manifolds

In this paper we suggest new effective criteria for the density property. This enables us to give a trivial proof of the original Anders\'en-Lempert result and to establish (almost free of charge) the algebraic density property for all linear algebraic groups whose connected components are different from tori or $\C_+$. As another application of this approach we tackle the question (asked among others by F. Forstneri\v{c}) about the density of algebraic vector fields on Euclidean space vanishing on a codimension 2 subvariety.

Projectivities between bundles of plane higher order curves

In [5] we give a theorem about projectivities between three line bundles in the projective plane with an almost trivial proof, and with a lot of special cases among which the theorems of Desargues, Pappus-Pascal, Pascal and many other lesser known results. In this paper we retake this idea, but now for projectivities between bundles of higher order curves in the plane.

Proof by computation in the [formula omitted] system

In informal mathematics, statements involving computations are seldom proved. Instead, it is assumed that readers of the proof can carry out the computations on their own. However, when using an automated proof development system based on type theory, the user is forced to find proofs for all claimed propositions, including computational statements. This paper presents a method to automatically prove statements from primitive recursive arithmetic. The method replaces logical formulas by boolean expressions. A correctness proof is constructed, which states that the original formula is derivable, if and only if the boolean expression equals true. Because the boolean expression reduces to true, the conversion rule yields a trivial proof of the equality. By combining this proof with the correctness proof, we get a proof for the original statement.

On Pták's derivation of the Jordan normal form

Some readers of [1] might appreciate the following comments that make more explicit how Ptak’s beautiful insight there leads to a trivial proof of (the basics of) the Jordan normal form. The proof of Theorem 1 of [1] can also be based on the observation that, X being finite-dimensional, the sequence {0} ⊆ ker A ⊆ ker A2 ⊆ · · · must eventually be stationary, i.e., ker A = ker Aq+p for some q and all p > 0. For such q, let Xr and Xs be the range and the kernel, respectively, of A , hence dim X = dim Xr + dim Xs . Further, for any x ∈ Xr ∩ Xs, x = Az for some z, and so z ∈ ker A2q = ker A , hence x = 0. Therefore, X is the direct sum of the two A-invariant subspaces Xs and Xr , and A is regular on Xr (since A is) and nilpotent on Xs . In the setup and notation of Theorem 2 of [1], there must be, by duality, some y0 in Y for which 〈x0Aq−1, y0〉 / = 0, hence the q-order matrix (〈x0Aj−1, y0A∗q−i〉: i,j = 1, . . . , q) is triangular with nonzero diagonal entries, therefore invertible, and this guarantees that X is the direct sum X0 + X′, with X0 the linear span of (x0Aj−1: j = 1, . . . , q) and X′ the annihilator of {y0A∗q−i : i = 1, . . . , q}, both of which are Ainvariant. Moreover, it shows (x0Aj−1: j = 1, . . . , q) to be a basis for X0, and the matrix representation, with respect to this basis, of A restricted to X0 has the familiar form of a Jordan block (for the eigenvalue 0). Now, X being finite-dimensional, there are A-invariant direct sum decompositions X = X1 + · · · + Xm that are minimal in the sense that none of its summands is the direct sum of two nontrivial A-invariant subspaces. Take any one such. Then the matrix representation for A with respect to any basis made up from bases for the summands Xi is block diagonal, with the ith block the matrix representation of the restriction Ai of A to Xi with respect to the chosen basis for Xi . Assuming the underlying field to be algebraically closed, the restriction Ai of A to Xi has some eigenvalue, λi , and, in view of the minimality of Xi , Theorem 1 ensures

On the N=2 superstring BRST operator

We show that the BRST charge for the N=2 superstring system can be written as Q= e −R( ∮ dz 2πi bγ +γ −) e R , where b and γ ± are super-reparametrizations ghosts. This provides a trivial proof of the nilpotence of this operator.

A note on the superstring BRST operator

We write the BRST operator of the N=1 superstring as Q=e −R( 1 2πi ∮ dz γ 2b)e R where γ and b are super-reparameterization ghosts. This provides a trivial proof that Q is nilpotent.

ENUMERATION OF CHORD DIAGRAMS AND AN UPPER BOUND FOR VASSILIEV INVARIANTS

We treat an enumeration problem of chord diagrams, which is shown to yield an upper bound for the dimension of the space of Vassiliev invariants for knots. We give an asymptotical estimate for this bound. As an aside, we present a trivial proof for the bound D!.

A note on calculating Condorcet probabilities

Probability voting models arrive at important conclusions on the paradox of voting. The computation of the probabilities uses the Selby relations for sums of powers of integers. The in- tegers are essentially the number of voters with linear preference orderings. With an ordered set of integers the equation implicit in these calculations may have to be amended. 1. Introduction Numerous articles have been devoted to computing probabilities for simple majority winners and the Condorcet In particular, the Gehrlein and Fishburn (1976) article which appeared in this journal has made a significant and well-cited contribution. The writing of this note has been suggested by vari- ous papers, in particular Berg (1985a, 1985b) and Lepelley (1986). This note is not directly concerned with the content of these articles but rather with an algebraic equation used in the method of computing probabilities for the Con- dorcet paradox, originating in the Gehrlein and Fishburn (1976) article. Working from the premise that studies of the paradox are based either on preference profiles or anonymous preference profiles (i.e., A-profiles), Gehrlein and Fishburn (1976: 2) comment that examination of the spaces of profiles and of A-profiles as they relate to Condorcet's paradox can provide new insights into the paradox. Assuming three alternatives or candidates and further assuming that the number of voters is odd, mathematical expressions are derived for the proportion of A-profiles that avoid the They (1976: 2) continue to show how the likelihood of the paradox can be computed on the basis of a probability distribution over A-profiles. The objective of this note is to examine an inequality derived from these probabilisfic models and to suggest that it may be amended when applied to an analysis of the Condorcet paradox in smaller committees, the quinntessen- tial example of the Condorcet The direction of the argument is an adaptation of a trivial proof in topology coupled with the identification of the number of voters as an ordered set of integers.

Perturbative renormalization of composite operators via flow equations II: Short distance expansion

We give a rigorous and very detailed derivation of the short distance expansion for a product of two arbitrary composite operators in the framework of the perturbative Euclidean massiveΦ 4 4 . The technically almost trivial proof rests on an extension of the differential flow equation method to Green functions with bilocal insertions, for which we also establish a set of generalized Zimmermann identities and Lowenstein rules.