Simple Proof Research Articles

A Short Simple Proof of Closedness of Convex Cones and Farkas’ Lemma

Proving that a finitely generated convex cone is closed is often considered the most difficult part of geometric proofs of Farkas’ lemma. We provide a short simple proof of this fact and (for completeness) derive Farkas’ lemma from it using well-known arguments.

New and simple proofs of Ramanujan’s modular equations of degree 11

On pp. 243 and 244 of his second notebook, Ramanujan recorded seven modular equations of degree 11 followed by two more which he had crossed out. The first modular equation in the list was proved by Berndt using theta function identities. The same was also proved by Venkatachaliengar in a different way. To the best of our knowledge, the only available proofs to the other six modular equations in the list are by Berndt. These proofs employ the theory of modular forms. Interestingly these are the first set of modular equations, the proofs to which by Berndt take a shift from the elementary algebraic methods to the theory of modular forms. In this paper, we reprove all these modular equations of degree 11 (except the first one in the list) by elementary algebraic techniques. First, we use two series identities due to Ye and Liu to prove one of the modular equations and its reciprocal. The remaining modular equations are proved by the method of parametrization.

Efficiency of any weighted geometric mean of the columns of a reciprocal matrix

We focus upon the reciprocal matrix component of the often discussed Analytic Hierarchy Process and its approximation by a consistent matrix formed from an efficient vector. It is known that a vector is efficient for a reciprocal matrix A if and only if a certain digraph is strongly connected. Here, we give a transparent and relatively simple matricial proof of this important result. Then we show that any Hadamard weighted geometric mean of the columns of a reciprocal matrix is efficient. This is a major generalization of the known result that the Hadamard geometric mean of all columns is efficient, and of the recent result that the geometric mean of any subset of the columns is efficient.

Gosper’s strange series: A new, simplified proof and generalizations

Gosper’s strange series: A new, simplified proof and generalizations

A Decomposition for Borel Measures \(\mu \le \mathcal{H}^{s}\)

We prove that every finite Borel measure \( \mu \) in \( \mathbb R^N \) that is bounded from above by the Hausdorff measure \( \mathcal{H}^s \) can be split in countable many parts \( \mu{\lfloor_{_{{E_k}}}} \) that are bounded from above by the Hausdorff content \( \mathcal{H}^s_\infty \). Such a result generalises a theorem due to R. Delaware that says that any Borel set with finite Hausdorff measure can be decomposed as a countable disjoint union of straight sets. We apply this decomposition to give a simpler proof for the existence of solutions of a Dirichlet problem involving an exponential nonlinearity.

A simple proof for the lower bound of the girth of graphs D(n,q)

The components of the graphs D(n,q) provide the best-known general lower bound for the number of edges in a graph with n vertices and no cycles of length less than g. In this paper, we give a new, short, and simpler proof of the fact that the length of the shortest cycle appearing in D(n,q) is n+5 when n is odd, and n+4 when n is even.

Lagrange Inversion Formula by Induction

We present a simple inductive proof of the Lagrange Inversion Formula.

A Simple Proof of ACC for Minimal Log Discrepancies for Surfaces

A Simple Proof of ACC for Minimal Log Discrepancies for Surfaces

Integral formula for quantum relative entropy implies data processing inequality

Integral representations of quantum relative entropy, and of the directional second and higher order derivatives of von Neumann entropy, are established, and used to give simple proofs of fundamental, known data processing inequalities: the Holevo bound on the quantity of information transmitted by a quantum communication channel, and, much more generally, the monotonicity of quantum relative entropy under trace-preserving positive linear maps – complete positivity of the map need not be assumed. The latter result was first proved by Müller-Hermes and Reeb, based on work of Beigi. For a simple application of such monotonicities, we consider any `divergence' that is non-increasing under quantum measurements, such as the concavity of von Neumann entropy, or various known quantum divergences. An elegant argument due to Hiai, Ohya, and Tsukada is used to show that the infimum of such a `divergence' on pairs of quantum states with prescribed trace distance is the same as the corresponding infimum on pairs of binary classical states. Applications of the new integral formulae to the general probabilistic model of information theory, and a related integral formula for the classical Rényi divergence, are also discussed.

Covariance-Matrix-Based Criteria for Network Entanglement.

Quantum networks offer a realistic and practical scheme for generating multiparticle entanglement and implementing multiparticle quantum communication protocols. However, the correlations that can be generated in networks with quantum sources and local operations are not yet well understood. Covariance matrices, which are powerful tools in entanglement theory, have been also applied to the network scenario. We present simple proofs for the decomposition of such matrices into the sum of positive semi-definite block matrices and, based on that, develop analytical and computable necessary criteria for preparing states in quantum networks. These criteria can be applied to networks where nodes share at most one source, such as all bipartite networks.

Successive coefficients for functions in the spirallike family

The aim of this paper is to compute the bounds of successive coefficients ||an+1|−|an|| for the family of univalent functions that are spirallike functions of non-negative order. This result not only improves a recent result but gives an alternate simple proof for the case of univalent spirallike functions.

A simple approach to Lieb–Thirring type inequalities

In [10] Nam proved a Lieb–Thirring Inequality for the kinetic energy of a fermionic quantum system, with almost optimal (semi-classical) constant and a gradient correction term. We present a stronger version of this inequality, with a much simplified proof. As a corollary we obtain a simple proof of the original Lieb–Thirring inequality.

A simplified proof of Serre’s conjecture

The purpose of the present article is to present a simplified proof of Serre’s modularity conjecture using the strong modularity lifting results currently available.

Sharp Coefficient Bounds for a Subclass of Bounded Turning Functions with a Cardioid Domain

In the present paper, we give a new simple proof on the sharp bounds of coefficient functionals related to the Carathéodory functions and make a correction on the extremal functions. The result is further used to investigate some initial coefficient bounds on a subclass of bounded turning functions R℘ associated with a cardioid domain. For functions in this class, we calculate the bounds of the Fekete–Szegö-type inequality and the second- and third-order Hankel determinants. All the results are proved to be sharp.

A structure theorem for streamed information

We identify the free half shuffle algebra of Schützenberger [31] with an algebra of real-valued functionals on paths, where the half shuffle emulates the integration of a functional against another. We then provide two, to our knowledge, new identities in arity 3 involving its commutator (area), and show that these are sufficient to recover the Zinbiel and Tortkara identities introduced by Dzhumadil'daev [11]. We then use these identities to provide a simple proof of the main result of Diehl et al. [8], namely that any element of the free half shuffle algebra can be expressed as a polynomial over iterated areas.Moreover, we consider minimal sets of Hall iterated integrals defined through the recursive application of the half shuffle product to Hall trees. Leveraging the duality between this set of Hall integrals and classical Hall bases of the free Lie algebra, we prove using combinatorial arguments that any element of the free half shuffle algebra can be written uniquely as a polynomial over Hall integrals. We interpret this result as a structure theorem for streamed information, loosely analogous to the unique prime factorisation of integers, allowing to split any real valued function on streamed data into two parts: a first that extracts and packages the streamed information into recursively defined atomic objects (Hall integrals), and a second that evaluates a polynomial function in these objects without further reference to the original stream. The question of whether a similar result holds if Hall integrals are replaced by Hall areas is left as an open conjecture.Finally, we construct a canonical, but to our knowledge, new decomposition of the free half shuffle algebra as shuffle power series in the greatest letter of the original alphabet with coefficients in a sub-algebra freely generated by a new alphabet with an infinite number of letters. We use this construction to provide a second proof of our structure theorem.

Conditional quantiles: An operator-theoretical approach

This paper derives several novel properties of conditional quantiles viewed as nonlinear operators. The results are organized in parallel to the usual properties of the expectation operator. We first define a τ-conditional quantile random set, relative to any sigma-algebra, as a set of solutions of an optimization problem. Then, well-known properties of unconditional quantiles, as translation invariance, comonotonicity, and equivariance to monotone transformations, are generalized to the conditional case. Moreover, a simple proof for Jensen’s inequality for conditional quantiles is provided. We also investigate continuity of conditional quantiles as operators with respect to different topologies and obtain a novel Fatou’s lemma for quantiles. Conditions for continuity in Lp and weak continuity are also derived. Then, the differentiability properties of quantiles are addressed. We demonstrate the validity of Leibniz’s rule for conditional quantiles for the cases of monotone, as well as separable functions. Finally, although the law of iterated quantiles does not hold in general, we characterize the maximum set of random variables for which this law holds, and investigate its consequences for the infinite composition of conditional quantiles.

PSPACE-Completeness of Reversible Deterministic Systems

We prove PSPACE-completeness of several reversible, fully deterministic systems. At the core, we develop a framework for such proofs (building on a result of Tsukiji and Hagiwara and a framework for motion planning through gadgets), showing that any system that can implement three basic gadgets is PSPACE-complete. We then apply this framework to four different systems, showing its versatility. First, we prove that Deterministic Constraint Logic is PSPACE-complete, fixing an error in a previous argument from 2008. Second, we give a new PSPACE-hardness proof for the reversible ‘billiard ball’ model of Fredkin and Toffoli from 40 years ago, newly establishing hardness when only two balls move at once. Third, we prove PSPACE-completeness of zero-player motion planning with any reversible deterministic interacting [Formula: see text]-tunnel gadget and a ‘rotate clockwise’ gadget (a zero-player analog of branching hallways). Fourth, we give simpler proofs that zero-player motion planning is PSPACE-complete with just a single gadget, the 3-spinner. These results should in turn make it even easier to prove PSPACE-hardness of other reversible deterministic systems.

On an instance of the small Cohen-Macaulay conjecture

We provide a simplified proof of a theorem proved by Tavanfar and Shimomoto which states that a quasi-Gorenstein deformation of a 3-dimensional quasi-Gorenstein local ring (A,m,k) with Hm2(A)=k admits a small Cohen-Macaulay module.

Positive First-order Logic on Words and Graphs

We study FO+, a fragment of first-order logic on finite words, where monadic predicates can only appear positively. We show that there is an FO-definable language that is monotone in monadic predicates but not definable in FO+. This provides a simple proof that Lyndon's preservation theorem fails on finite structures. We lift this example language to finite graphs, thereby providing a new result of independent interest for FO-definable graph classes: negation might be needed even when the class is closed under addition of edges. We finally show that the problem of whether a given regular language of finite words is definable in FO+ is undecidable.

Long range order for random field Ising and Potts models

AbstractWe present a new and simple proof for the classic results of Imbrie (1985) and Bricmont–Kupiainen (1988) that for the random field Ising model in dimension three and above there is long range order at low temperatures with presence of weak disorder. With the same method, we obtain a couple of new results: (1) we prove that long range order exists for the random field Potts model at low temperatures with presence of weak disorder in dimension three and above; (2) we obtain a lower bound on the correlation length for the random field Ising model at low temperatures in dimension two (which matches the upper bound in Ding–Wirth (2020)). Our proof is based on an extension of the Peierls argument with inputs from Chalker (1983), Fisher–Fröhlich–Spencer (1984), Ding–Wirth (2020) and Talagrand's majorizing measure theory (1980s) (and in particular, our proof does not involve the renormalization group theory).