Simple Proof Research Articles

Simple proofs of certain inequalities with logarithmic coefficients of univalent functions

In this paper, we give simple proofs for the bounds (some of them sharp) of the difference of the moduli of the second and the first logarithmic coefficient for the general class of univalent functions and for the class of convex univalent functions.

A Simple Proof of the Nonuniform Kahn–Kalai Conjecture

A Simple Proof of the Nonuniform Kahn–Kalai Conjecture

Unimodal sequences: From Isaac Newton to June Huh

In this paper, we present a simple proof of a famous result of Newton on unimodal sequences associated to polynomials with real roots. We then give a brief exposition of the recent work of Huh on such sequences arising from combinatorial geometries.

Efficient Federated Low Rank Matrix Recovery via Alternating GD and Minimization: A Simple Proof

Efficient Federated Low Rank Matrix Recovery via Alternating GD and Minimization: A Simple Proof

Twin-width of planar graphs; a short proof

The fascinating question of the maximum value of twin-width on planar graphs is nowadays not far from the final resolution; there is a lower bound of 7 coming from a construction by Král’ and Lamaison (2022), and an upper bound of 8 by Hliněný and Jedelský (2022). The upper bound (currently best) of 8, however, is rather complicated and involved. In the paper we give a short and simple self-contained proof that the twin-width of planar graphs is at most 11. We believe that this short proof can also shed more light on the topic of upper bound(s) on the twin-width of planar and beyond-planar graphs in general.

Simple proof that there is no sign problem in path integral Monte Carlo simulations of fermions in one dimension.

It is widely known that there is no sign problem in path integral Monte Carlo (PIMC) simulations of fermions in one dimension. As far as the author is aware, there is no direct proof of this in the literature. This work shows that the sign of the N-fermion antisymmetric free propagator is given by the product of all possible pairs of particle separations, or relative displacements. For a nonvanishing closed-loop product of such propagators, as required by PIMC, all relative displacements from adjacent propagators are paired into perfect squares, and therefore the loop product must be positive, but only in one dimension. By comparison, permutation sampling, which does not evaluate the determinant of the antisymmetric propagator exactly, remains plagued by a low-level sign problem, even in one dimension.

Some progress on the Aharoni–Korman conjecture

Aharoni and Korman (1992) [3] conjectured that every ordered set with no infinite antichains possesses a chain and a partition into antichains so that each part intersects the chain.The conjecture is verified for posets whose incomparability graph is locally finite. It follows that the conjecture is true for (3+1)-free posets with no infinite antichains.We give a necessary and sufficient condition for a P4-free graph to be a cograph. This allows us to obtain a simple proof of the fact that finite P4-free graphs are finite cographs. We also prove that N-free chain complete posets and N-free posets with no infinite antichains are series-parallel.As a consequence, we obtain that the conjecture is true for N-free posets with no infinite antichain.

On some algorithms for estimation in Gaussian graphical models

Abstract In Gaussian graphical models, the likelihood equations must typically be solved iteratively. This paper investigates two algorithms: a version of iterative proportional scaling, which avoids inversion of large matrices, and an algorithm based on convex duality and operating on the covariance matrix by neighbourhood coordinate descent, which corresponds to the graphical lasso with zero penalty. For large, sparse graphs, the iterative proportional scaling algorithm appears feasible and has simple convergence properties. The algorithm based on neighbourhood coordinate descent is extremely fast and less dependent on sparsity, but needs a positive-definite starting value to converge. We provide an algorithm for finding such a starting value for graphs with low colouring number. As a consequence, we also obtain a simplified proof of existence of the maximum likelihood estimator in such cases.

On some extensions of the Hua determinant inequality and a question of Poon

In this note, we give simple proofs of two recent extensions of the Hua determinant inequality. We also answer a question of Poon affirmatively on a Young type inequality, which appears in his investigation of inequalities on eigenvalues arising from the Hua determinant inequality.

Chromatic fixed point theory and the Balmer spectrum for extraspecial 2-groups

abstract: In the early 1940s, P. A. Smith showed that if a finite $p$-group $G$ acts on a finite dimensional complex $X$ that is mod $p$ acyclic, then its space of fixed points, $X^G$, will also be mod $p$ acyclic. In their recent study of the Balmer spectrum of equivariant stable homotopy theory, Balmer and Sanders were led to study a question that can be shown to be equivalent to the following: if a $G$-space $X$ is a equivariant homotopy retract of the $p$-localization of a based finite $G$-C.W. complex, given $H<G$ and $n$, what is the smallest $r$ such that if $X^H$ is acyclic in the $(n+r)$th Morava $K$-theory, then $X^G$ must be acyclic in the $n$th Morava $K$-theory? Barthel et.~al. then answered this when $G$ is abelian, by finding general lower and upper bounds for these ``blue shift'' numbers which agree in the abelian case. In our paper, we first prove that these potential chromatic versions of Smith's theorem are equivalent to chromatic versions of a 1952 theorem of E. E. Floyd, which replaces acyclicity by bounds on dimensions of mod $p$ homology, and thus applies to all finite dimensional $G$-spaces. This unlocks new techniques and applications in chromatic fixed point theory. Applied to the problem of understanding blue shift numbers, we are able to use classic constructions and representation theory to search for lower bounds. We give a simple new proof of the known lower bound theorem, and then get the first results about nonabelian 2-groups that do not follow from previously known results. In particular, we are able to determine all blue shift numbers for extraspecial 2-groups. Applied in ways analogous to Smith's original applications, we prove new fixed point theorems for $K(n)_*$-homology disks and spheres. Finally, our methods offer a new way of using equivariant results to show the collapsing of certain Atiyah-Hirzebruch spectral sequences in certain cases. Our criterion appears to apply to the calculation of all 2-primary Morava $K$-theories of all real Grassmanians.

Using intrusive approaches as a step towards accounting for stochasticity in wind turbine design

Current wind turbine design methods require tens of thousands of time-domain simulations and use different random seeds to account for the stochasticity of the environmental conditions. The account of stochasticity is nonintrusive because the sampling method calls a deterministic model multiple times without changing its underlying equations. In this work, we investigate and demonstrate using simple proof of concepts how intrusive approaches can be used to directly account for stochasticity in the equations representing a mechanical system. Our long term goal is to apply such methodology to the design of wind turbines without requiring an excessive number of simulations. Intrusive methods manipulate stochastic variables directly to provide the probability density functions (PDFs) of the states and outputs at any time as functions of the PDFs of the inputs. We illustrate how different methods can be used with a reduced-order model of a wind turbine with one degree of freedom and for linear and nonlinear models. We discuss how the methods can be extended and what it will take to apply them to a level of fidelity similar to current state-of-the-art wind turbine design tools.

Todd Polynomials and Hirzebruch Numbers

In 1956 Hirzebruch found an explicit formula for the denominators of the Todd polynomials, which was proved later in his joint work with Atiyah. We present a new formula for the Todd polynomials in terms of the “forgotten symmetric functions,” which follows from our previous work on complex cobordisms. In particular, this leads to a simpler proof of the Hirzebruch formula and provides new interpretations for the Hirzebruch numbers.

Coarse Geometry of Quasi-Transitive Graphs Beyond Planarity

We study geometric and topological properties of infinite graphs that are quasi-isometric to a planar graph of bounded degree. We prove that every locally finite quasi-transitive graph excluding a minor is quasi-isometric to a planar graph of bounded degree. We use the result to give a simple proof of the result that finitely generated minor-excluded groups have Assouad-Nagata dimension at most 2 (this is known to hold in greater generality, but all known proofs use significantly deeper tools). We also prove that every locally finite quasi-transitive graph that is quasi-isometric to a planar graph is $k$-planar for some $k$ (i.e. it has a planar drawing with at most $k$ crossings per edge), and discuss a possible approach to prove the converse statement.

Counting spanning subgraphs in dense hypergraphs

AbstractWe give a simple method to estimate the number of distinct copies of some classes of spanning subgraphs in hypergraphs with a high minimum degree. In particular, for each $k\geq 2$ and $1\leq \ell \leq k-1$ , we show that every $k$ -graph on $n$ vertices with minimum codegree at least \begin{equation*} \left \{\begin {array}{l@{\quad}l} \left (\dfrac {1}{2}+o(1)\right )n & \text { if }(k-\ell )\mid k,\\[5pt] \left (\dfrac {1}{\lceil \frac {k}{k-\ell }\rceil (k-\ell )}+o(1)\right )n & \text { if }(k-\ell )\nmid k, \end {array} \right . \end{equation*} contains $\exp\!(n\log n-\Theta (n))$ Hamilton $\ell$ -cycles as long as $(k-\ell )\mid n$ . When $(k-\ell )\mid k$ , this gives a simple proof of a result of Glock, Gould, Joos, Kühn, and Osthus, while when $(k-\ell )\nmid k$ , this gives a weaker count than that given by Ferber, Hardiman, and Mond, or when $\ell \lt k/2$ , by Ferber, Krivelevich, and Sudakov, but one that holds for an asymptotically optimal minimum codegree bound.

A note on the Jacobson radical of Ore extensions

Let R be a ring with a monomorphism α and an α-derivation δ. In this article, we give a simple and different proof about the semiprimitivity of Ore extensions which states that the skew polynomial ring R [ x ; α , δ ] is semiprimitive reduced if and only if R is α-rigid. This unifies and extends a number of known results on the Jacobson radical in the special cases. Also, as an application of our results, by imposing constraints on α and δ, we completely identify the Jacobson radical of rings whose the set of all nilpotent elements has special conditions. Important examples are provided to illustrate the applications of the results.

Twisted Kähler–Einstein metrics in big classes

AbstractWe prove existence of twisted Kähler–Einstein metrics in big cohomology classes, using a divisorial stability condition. In particular, when is big, we obtain a uniform Yau–Tian–Donaldson (YTD) existence theorem for Kähler–Einstein (KE) metrics. To achieve this, we build up from scratch the theory of Fujita–Odaka type delta invariants in the transcendental big setting, using pluripotential theory. We do not use the K‐energy in our arguments, and our techniques provide a simple roadmap to prove YTD existence theorems for KE type metrics, that only needs convexity of the appropriate Ding energy. As an application, we give a simplified proof of Li–Tian–Wang's existence theorem in the log Fano setting.

On linear combinations of binomial OWA functions

We consider the binomial decomposition of ordered weighted averaging (OWA) functions proposed by Calvo and De Baets (1998) in the framework of Choquet integration. Our aim in the paper is to further investigate the equivalence between the two representations of OWA functions involved in the binomial decomposition: the usual canonical representation in terms of the order statistics, and the binomial representation in terms of the binomial OWA functions. We describe and discuss in detail the linear transformations that relate the coefficients of these two equivalent representations: the original expression of the weights in terms of the coefficients of the binomial representation, and its inverse, the expression of those coefficients in terms of the weights. In both cases simple and direct proofs are presented. Moreover, we consider the linear transformations between the two representations in the general linear algebra framework of unconstrained linear combinations of binomial OWA functions. In this perspective we obtain compact matrix expressions for the linear transformations, which also offer new insight on the geometry of the coefficient constraints in the binomial representation of OWA functions.

Analytic representations of backward shifts on weighted Bergman and Dirichlet spaces

The backward shifts (the adjoints of multiplication by z) on weighted Bergman spaces of the unit disk are important in operator theory. For example, Agler [1] shows their restrictions to invariant subspaces of vector-valued weighted Bergman spaces are model operators for hypercontractions. We present analytic representations of backward shifts on weighted Bergman spaces and Dirichlet type spaces. Since these analytic representations are not space specific, it is interesting to study these operators on Hilbert spaces of holomorprhic functions on the unit disk. We initiate this study by developing several properties of these operators. In particular, we obtain simple proofs and extend the invariant subspace theorem of shift plus Volterra operator on the Hardy space by Čučković and Paudyal [16]. Furthermore, we describe invariant subspaces of shift plus Volterra operator on the Bergman space by relating them to the invariant subspaces of the Dirichlet shift. Analytic representations of the tuple of backward shifts on weighted Bergman spaces of the unit ball are also discussed.

Why the Bethe-West-Yennie formula for Coulomb-nuclear interference is inconsistent

We give a new and simple proof of the inconsistency of the Bethe-West-Yennie parametrization for Coulomb-nuclear interference.

Optimal Error Bounds in the Absence of Constraint Qualifications with Applications to p-Cones and Beyond

We prove tight Hölderian error bounds for all p-cones. Surprisingly, the exponents differ in several ways from those that have been previously conjectured. Moreover, they illuminate p-cones as a curious example of a class of objects that possess properties in three dimensions that they do not in four or more. Using our error bounds, we analyse least squares problems with p-norm regularization, where our results enable us to compute the corresponding Kurdyka–Łojasiewicz exponents for previously inaccessible values of p. Another application is a (relatively) simple proof that most p-cones are neither self-dual nor homogeneous. Our error bounds are obtained under the framework of facial residual functions, and we expand it by establishing for general cones an optimality criterion under which the resulting error bound must be tight. Funding: The second author was partly supported by the Japan Society for the Promotion of Science Grant-in-Aid for Early-Career Scientists [Grants 19K20217, 23K16844] and Grant-in-Aid for Scientific Research [Grant (B)21H03398]. The third author was partly supported by the Hong Kong Research Grants Council [Grant PolyU153000/20p].