Simple Proof Research Articles

Tight bounds for antidistinguishability and circulant sets of pure quantum states

A set of pure quantum states is said to be antidistinguishable if upon sampling one at random, there exists a measurement to perfectly determine some state that was not sampled. We show that antidistinguishability of a set of n pure states is equivalent to a property of its Gram matrix called (n−1)-incoherence, thus establishing a connection with quantum resource theories that lets us apply a wide variety of new tools to antidistinguishability. As a particular application of our result, we present an explicit formula (not involving any semidefinite programming) that determines whether or not a set with a circulant Gram matrix is antidistinguishable. We also show that if all inner products are smaller than (n−2)/(2n−2) then the set must be antidistinguishable, and we show that this bound is tight when n≤4. We also give a simpler proof that if all the inner products are strictly larger than (n−2)/(n−1), then the set cannot be antidistinguishable, and we show that this bound is tight for all n.

New Proofs of Fixed Point Theorems on Quasi-metric Spaces

Based on our 2023 Metatheorem in Ordered Fixed Point Theory, we give very simple proofs of known fixed point theorems for various multimap classes on quasi-metric spaces. Such classes are represented by the Banach contractions, the Rus-Hicks-Rhoades maps, the Nadler multimaps, Covitz-Nadler multimaps, and others. Consequently, we obtain simple proofs of a large number of known theorems on extremal elements, fixed points, stationary points for several classes of maps or multimaps. Finally, we add some known theorems for which our Metatheorem does not work.

Fundamental polyhedra of projective elementary groups

Abstract For 𝓞 an imaginary quadratic ring, we compute a fundamental polyhedron in hyperbolic 3-space for the action of PE2(𝓞), the projective elementary subgroup of PSL2(𝓞). This allows for new, simplified proofs of theorems of Cohn, Nica, Fine, and Frohman. Namely, we obtain a presentation for PE2(𝓞), show that it has infinite index and is its own normalizer in PSL2(𝓞), and split PSL2(𝓞) into a free product with amalgamation that has PE2(𝓞) as one of its factors.

On the existence of solutions of the system of nonlinear quasi-mixed equilibrium problems and their stability of the procedure scheme

In this paper, we provide a simplified proof of the existence of solutions for the system of nonlinear quasi-mixed equilibrium problems studied by Suantai and Petrot (Appl. Math. Lett. 24:308–313, 2011), utilizing Banach’s fixed point theorem. This result remains valid under weaker assumptions through the product space approach. Moreover, we show that the stability analysis of the iterative algorithm proposed in their work (and in (J. Inequal. Appl. 2010:437976, 2010)) contains a flaw, which we address using Ostrowski’s classical result from 1967 (Z. Angew. Math. Mech. 47:77–81, 1967). Finally, we examine and improve the convergence theorem for solving a system of variational inequalities as established by Chang et al. (Appl. Math. Lett. 20:329–334, 2007).

How Many Homeomorphisms are “Many Homeomorphisms”?

Motivated by the study of discrete dynamics, we present a simple and self-contained proof that the space of all the homeomorphisms of a compact metric space is large, in the sense of the Baire categories.

Compactifications of pseudofinite and pseudoamenable groups

We first give simplified and corrected accounts of some results in work by Pillay (2017) on compactifications of pseudofinite groups. For instance, we use a classical theorem of Turing (1938) to give a simplified proof that any definable compactification of a pseudofinite group has an abelian connected component. We then discuss the relationship between Turing’s work, the Jordan–Schur theorem, and a (relatively) more recent result of Kazhdan (1982) on approximate homomorphisms, and we use this to widen our scope from finite groups to amenable groups. In particular, we develop a suitable continuous logic framework for dealing with definable homomorphisms from pseudoamenable groups to compact Lie groups. Together with the stabilizer theorems of Hrushovski (2012) and Montenegro et al. (2020), we obtain a uniform (but non-quantitative) analogue of Bogolyubov’s lemma for sets of positive measure in discrete amenable groups. We conclude with brief remarks on the case of amenable topological groups.

On Decidable and Undecidable Extensions of Simply Typed Lambda Calculus

The decidability of the reachability problem for finitary PCF has been used as a theoretical basis for fully automated verification tools for functional programs. The reachability problem, however, often becomes undecidable for a slight extension of finitary PCF with side effects, such as exceptions, algebraic effects, and references, which hindered the extension of the above verification tools for supporting functional programs with side effects. In this paper, we first give simple proofs of the undecidability of four extensions of finitary PCF, which would help us understand and analyze the source of undecidability. We then focus on an extension with references, and give a decidable fragment using a type system. To our knowledge, this is the first non-trivial decidable fragment that features higher-order recursive functions containing reference cells.

Calculational Design of Hyperlogics by Abstract Interpretation

We design various logics for proving hyper properties of iterative programs by application of abstract interpretation principles. In part I, we design a generic, structural, fixpoint abstract interpreter parameterized by an algebraic abstract domain describing finite and infinite computations that can be instantiated for various operational, denotational, or relational program semantics. Considering semantics as program properties, we define a post algebraic transformer for execution properties (e.g. sets of traces) and a Post algebraic transformer for semantic (hyper) properties (e.g. sets of sets of traces), we provide corresponding calculuses as instances of the generic abstract interpreter, and we derive under and over approximation hyperlogics. In part II, we define exact and approximate semantic abstractions, and show that they preserve the mathematical structure of the algebraic semantics, the collecting semantics post, the hyper collecting semantics Post, and the hyperlogics. Since proofs by sound and complete hyperlogics require an exact characterization of the program semantics within the proof, we consider in part III abstractions of the (hyper) semantic properties that yield simplified proof rules. These abstractions include the join, the homomorphic, the elimination, the principal ideal, the order ideal, the frontier order ideal, and the chain limit algebraic abstractions, as well as their combinations, that lead to new algebraic generalizations of hyperlogics, including the ∀∃ ∗ , ∀∀ ∗ , and ∃∀ ∗ hyperlogics.

Rational approximation of holomorphic semigroups revisited

AbstractUsing a recently developed ‐calculus we propose a unified approach to the study of rational approximations of holomorphic semigroups on Banach spaces. We provide unified and simple proofs to a number of basic results on semigroup approximations and substantially improve some of them. We show that many of our estimates are essentially optimal, thus complementing the existing literature.

Nash equilibrium and minimax theorems via variational tools of convex analysis

In this paper, we first provide a simple variational proof of the existence of Nash equilibrium in Hilbert spaces by using optimality conditions in convex minimization and Schauder's fixed-point theorem. Then applications of convex analysis and generalized differentiation are given to the existence of Nash equilibrium and extended versions of von Neumann's minimax theorem in locally convex topological vector spaces. Our analysis in this part combines generalized differential tools of convex analysis with elements of fixed point theory revolving around Kakutani's fixed-point theorem and related issues.

On a Holm-related MTP for rejecting at least k hypotheses: general validity, optimality property, confidence regions, and applications

ABSTRACT This article concerns p-value-based multiple testing procedures (MTPs) that can be used in a confirmatory clinical study under minimal assumptions in case the requirement for study-success is that at least k out of m primary/important hypotheses become rejected. Recently, a simple, generally valid Holm-type MTP was discussed that can be used for such a requirement for any k from one to m. It can only reject at least k (or zero) hypotheses, but this increases the power to reject k or more hypotheses compared to Holm’s step-down MTP. The present article provides a simple formulation and proof of strong family-wise error rate (FWER) control for a stepwise MTP that is sharper in that for any k strictly between one and m it: (a) always rejects at least as much, and (b) can potentially reject fewer than k hypotheses. This sharper MTP too is generally valid, without any assumption about logical or stochastic relationships. It has a gatekeeping step, followed by m steps where ordered primary p-values are compared to critical constants and rejections are made in a step-down manner. These constants have the optimality property that under a natural monotonicity restriction, they cannot be increased without losing the general strong FWER control. Confidence regions like those for Holm’s MTP are provided. Applications are discussed in connection with three interesting approaches proposed earlier for confirmatory studies: (a) the Superiority-Noninferiority approach; (b) Fallback tests for co-primary endpoints; and (c) Multistage gatekeeping MTPs that utilize so-called k-truncated Holm MTPs in some stages.

The sparse circular law, revisited

AbstractLet be an matrix with iid entries distributed as Bernoulli random variables with parameter . Rudelson and Tikhomirov, in a beautiful and celebrated paper, show that the distribution of eigenvalues of is approximately uniform on the unit disk as as long as , which is the natural necessary condition. In this paper, we give a much simpler proof of this result, in its full generality, using a perspective we developed in our recent proof of the existence of the limiting spectral law when is bounded. One feature of our proof is that it entirely avoids the use of ‐nets and, instead, proceeds by studying the evolution of the singular values of the shifted matrices as we incrementally expose the randomness in the matrix.

On Quantified Splitting Proof System for Propositional Calculi

In this paper, some new quantified propositional proof system is introduced and compared by proof complexities with other quantified and not quantified propositional proof systems. It is proved that the introduced system 1) is polynomially equivalent to its quantifier-free variant and 2) has exponential speed-up by sizes over some variants of the quantified resolution system. As the introduced system has a very simple proof construction strategy, it can be very useful not only in Logic, and therefore in Artificial Intelligence, but also in areas such as Computational Biology and Medical Diagnosis.

A simple proof of Blackwell’s theorem on the comparison of experiments for a general state space

A simple proof of Blackwell’s theorem on the comparison of experiments for a general state space

Legal Uncertainty in the Application of Simple Proof in Bankruptcy Cases and Suspension of Debt Payment Obligations

Legal Uncertainty in the Application of Simple Proof in Bankruptcy Cases and Suspension of Debt Payment Obligations

Strong Greedoid Structure of $r$-Removed $P$-Orderings

Inspired by the notion of $r$-removed $P$-orderings introduced in the setting of Dedekind domains by Bhargava, we generalize it to the framework of arbitrary ultrametric spaces. We show that sets of maximal "$r$-removed perimeter" can be constructed by a greedy algorithm and form a strong greedoid. This gives a simplified proof of several theorems previously obtained by Bhargava, as well as generalises some results of Grinberg and Petrov who considered the case $r=0$ corresponding, in turn, to simple $P$-orderings.

Local complete intersections and Weierstrass points

This work presents a simple proof that the moduli space of complete integral Gorenstein curves with a prescribed symmetric Weierstrass semigroup becomes a weighted projective space, even for fields of positive characteristic, when the associated monomial curve is a local complete intersection.

From Condorcet’s paradox to Arrow: yet another simple proof of the impossibility theorem

From Condorcet’s paradox to Arrow: yet another simple proof of the impossibility theorem

The Fourier transform on valuations is the Fourier transform

Alesker has proved the existence of a remarkable isomorphism of the space of translation-invariant smooth valuations that has the same functorial properties as the classical Fourier transform. In this paper, we show how to directly describe this isomorphism in terms of the Fourier transform on functions. As a consequence, we obtain simple proofs of the main properties of the Alesker–Fourier transform. One of these properties was previously only conjectured by Alesker and is proved here for the first time.

Strongly hyperbolic quasilinear systems revisited, with applications to relativistic fluid dynamics

We revisit the theory of first-order quasilinear systems with diagonalizable principal part and only real eigenvalues, what is commonly referred to as strongly hyperbolic systems. We provide a self-contained and simple proof of local well-posedness, in the Hadamard sense, of the Cauchy problem. Our regularity assumptions are very minimal. As an application, we apply our results to systems of ideal and viscous relativistic fluids, where the theory of strongly hyperbolic equations has been systematically used to study several systems of physical interest.