Simple Proof Research Articles

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].

A short proof of Haemers' conjecture on the Seidel energy of graphs

Haemers' conjecture on the Seidel energy of graphs, states that the energy of the Seidel matrix of any graph of order n is at least 2n−2, where the equality holds for the complete graph. In this paper, we give a short simple proof of this conjecture.

Toward robust and high-throughput detection of seed defects in X-ray images via deep learning

BackgroundThe detection of internal defects in seeds via non-destructive imaging techniques is a topic of high interest to optimize the quality of seed lots. In this context, X-ray imaging is especially suited. Recent studies have shown the feasibility of defect detection via deep learning models in 3D tomography images. We demonstrate the possibility of performing such deep learning-based analysis on 2D X-ray radiography for a faster yet robust method via the X-Robustifier pipeline proposed in this article.Results2D X-ray images of both defective and defect-free seeds were acquired. A deep learning model based on state-of-the-art object detection neural networks is proposed. Specific data augmentation techniques are introduced to compensate for the low ratio of defects and increase the robustness to variation of the physical parameters of the X-ray imaging systems. The seed defects were accurately detected (F1-score >90%), surpassing human performance in computation time and error rates. The robustness of these models against the principal distortions commonly found in actual agro-industrial conditions is demonstrated, in particular, the robustness to physical noise, dimensionality reduction and the presence of seed coating.ConclusionThis work provides a full pipeline to automatically detect common defects in seeds via 2D X-ray imaging. The method is illustrated on sugar beet and faba bean and could be efficiently extended to other species via the proposed generic X-ray data processing approach (X-Robustifier). Beyond a simple proof of feasibility, this constitutes important results toward the effective use in the routine of deep learning-based automatic detection of seed defects.

Bogomol’nyi-like equations in gravity theories

Using the Bogomol’nyi–Prasad–Sommerfield Lagrangian method, we show that gravity theory coupled to matter in various dimensions may possess Bogomol’nyi-like equations, which are first-order differential equations, satisfying the Einstein equations and the Euler–Lagrange equations of classical fields (U(1) gauge and scalar fields). In particular we consider static and spherically symmetric solutions by taking proper ansatzes and then we find an effective Lagrangian density that can reproduce the Einstein equations and the Euler–Lagrange equations of the classical fields. We consider the BPS Lagrangian density to be linear function of first-order derivative of all the fields. From these two Lagrangian desities we are able to obtain the Bogomol’nyi-like equations whose some of solutions are well-known such as Schwarzschild, Reissner–Nordström, Tangherlini black holes, and the recent black holes with scalar hair in three dimensions (Phys. Rev. D 107, 124047). Using these Bogomol’nyi-like equations, we are also able to find new solutions for scalar hair black holes in three and four dimensional spacetime. Furthermore we show that the Bogomol’nyi–Prasad–Sommerfield Lagrangian method can provide a simple alternative proof of black holes uniqueness theorems in any dimension.

Simple proofs for lattice coverings and sparse tensors

We provide simple proofs of analogues for coverings numbers of lattices of several recently studied basic statements on the ranks of tensors. These analogues involve in particular the interplay between different covering numbers, the existence of sublattices with high covering number and satisfying various conditions, and upper bounds on the size of the set of possible coverings with minimal size. We highlight the differences and analogies between the proofs in both settings.

On eigenvalues of certain special matrices

Let g : R → [ 0 , ∞ ) be a conditionally negative definite function and f : [ 0 , ∞ ) → [ 0 , ∞ ) be a Bernstein function. We prove that the function h = f ∘ g is conditionally negative definite and that for distinct real numbers p 1 , p 2 , … , p n the inertia of the matrix [ h ( p i − p j ) ] is ( 1 , 0 , n − 1 ) if f is non-linear and g ( x ) = 0 only for x = 0. We also present a new and simple proof to show that the matrix [ log ⁡ ( 1 − p i p j ) ] is negative definite for n distinct positive real numbers p i < 1 , ∀ i . These results supplement and unify earlier results proved by a number of authors including Dyn, Goodman and Michelli, Bhatia and Jain, Garg and Aujla, Garg and Agarwal for operator monotone functions.

Scenario-Based Proofs for Concurrent Objects

Concurrent objects form the foundation of many applications that exploit multicore architectures and their importance has lead to informal correctness arguments, as well as formal proof systems. Correctness arguments (as found in the distributed computing literature) give intuitive descriptions of a few canonical executions or "scenarios" often each with only a few threads, yet it remains unknown as to whether these intuitive arguments have a formal grounding and extend to arbitrary interleavings over unboundedly many threads. We present a novel proof technique for concurrent objects, based around identifying a small set of scenarios (representative, canonical interleavings), formalized as the commutativity quotient of a concurrent object. We next give an expression language for defining abstractions of the quotient in the form of regular or context-free languages that enable simple proofs of linearizability. These quotient expressions organize unbounded interleavings into a form more amenable to reasoning and make explicit the relationship between implementation-level contention/interference and ADT-level transitions. We evaluate our work on numerous non-trivial concurrent objects from the literature (including the Michael-Scott queue, Elimination stack, SLS reservation queue, RDCSS and Herlihy-Wing queue). We show that quotients capture the diverse features/complexities of these algorithms, can be used even when linearization points are not straight-forward, correspond to original authors' correctness arguments, and provide some new scenario-based arguments. Finally, we show that discovery of some object's quotients reduces to two-thread reasoning and give an implementation that can derive candidate quotients expressions from source code.

Effectiveness of Law Enforcement on Corporate Bankruptcy Status

&lt;p&gt;This study aims to evaluate the effectiveness of law enforcement on the status of bankruptcy in Indonesia. Lawrence M. Friedman's legal system theory is used because it has a comprehensive scope to evaluate the effectiveness of enforcing legislation. This study uses juridical-normative research with statutory and case approach. The laws and regulations studied are Law no. 37 of 2004 concerning Bankruptcy and Postponement of Debt Payment Obligations (KPKPU), while the case study takes the case of the bankruptcy of Telkomsel in 2012. This study finds that bankruptcy law enforcement in Indonesia has not been effective. This is because Indonesian bankruptcy law still has weaknesses in terms of substance, structure, and legal culture. In addition, this study finds that the fundamental weakness of Law no. 37 of 2004 is the application of simple proof as a mechanism for imposing bankruptcy statements to debtors. The application of this simple evidence makes law enforcers (judges) tend to ignore facts other than the two conditions stipulated in Law no. 37 of 2004 to impose bankruptcy status, namely the existence of two or more creditors and the existence of one debt that is due and collectible. In the end, the simple evidence mechanism does not open up opportunities for law enforcement officials to assess the debtor's ability to pay off their debts.&lt;/p&gt;

A note on 𝐶^{1,𝛼}-smooth approximation of Lipschitz functions

We show that on super-reflexive spaces a Moreau-Yosida type of regularisation by infimal convolution together with a known insertion-type theorem (a variant of Ilmanen’s lemma) easily give an approximation of a Lipschitz function by a C 1 , α C^{1,\alpha } -smooth Lipschitz function with the same Lipschitz constant. This is a generalisation of the well-known theorem of J.-M. Lasry and P.-L. Lions from Hilbert spaces. It also gives a new self-contained and probably simpler proof of the Lasry-Lions theorem.

A variational proof of a disentanglement theorem for multilinear norm inequalities

The basic disentanglement theorem established by the present authors states that estimates on a weighted geometric mean over (convex) families of functions can be disentangled into quantitatively linked estimates on each family separately. On the one hand, the theorem gives a uniform approach to classical results including Maurey's factorisation theorem and Lozanovskiĭ's factorisation theorem, and, on the other hand, it underpins the duality theory for multilinear norm inequalities developed in our previous two papers.In this paper we give a simple proof of this basic disentanglement theorem. Whereas the approach of our previous paper was rather involved – it relied on the use of minimax theory together with weak*-compactness arguments in the space of finitely additive measures, and an application of the Yosida–Hewitt theory of such measures – the alternate approach of this paper is rather straightforward: it instead depends upon elementary perturbation and compactness arguments.

On Fall-Colorable Graphs

A fall k-coloring of a graph G is a proper k-coloring of G such that each vertex has at least one neighbor in each of the other color classes. A graph G which has a fall k-coloring is equivalent to having a partition of the vertex set V(G) in k independent dominating sets. In this paper, we first prove that for any fall k-colorable graph G with order n, the number of edges of G is at least (n(k−1)+r(k−r))/2, where r≡n(modk) and 0≤r≤k−1, and the bound is tight. Then, we obtain that if G is k-colorable (k≥2) and the minimum degree of G is at least k−2k−1n, then G is fall k-colorable and this condition of minimum degree is the best possible. Moreover, we give a simple proof for an NP-hard result of determining whether a graph is fall k-colorable, where k≥3. Finally, we show that there exist an infinite family of fall k-colorable planar graphs for k∈{5,6}.

Hyperstability of Cauchy and Jensen functional equations in 2-normed spaces

In this article, with simple and short proofs without applying fixed point theorems, some hyperstability results corresponding to the functional equations of Cauchy and Jensen are presented in 2-normed spaces. We also obtain some results on hyperstability for the general linear functional equation f(ax+by)=Af(x)+Bf(y)+C\\documentclass[12pt]{minimal} \\usepackage{amsmath} \\usepackage{wasysym} \\usepackage{amsfonts} \\usepackage{amssymb} \\usepackage{amsbsy} \\usepackage{mathrsfs} \\usepackage{upgreek} \\setlength{\\oddsidemargin}{-69pt} \\begin{document}$f(ax+by)=Af(x)+Bf(y)+C$\\end{document}.

Connected greedy colourings of perfect graphs and other classes: the good, the bad and the ugly

The Grundy number of a graph is the maximum number of colours used by the "First-Fit" greedy colouring algorithm over all vertex orderings. Given a vertex ordering $\sigma= v_1,\dots,v_n$, the "First-Fit" greedy colouring algorithm colours the vertices in the order of $\sigma$ by assigning to each vertex the smallest colour unused in its neighbourhood. By restricting this procedure to vertex orderings that are connected, we obtain {\em connected greedy colourings}. For some graphs, all connected greedy colourings use exactly $\chi(G)$ colours; they are called {\em good graphs}. On the opposite, some graphs do not admit any connected greedy colouring using only $\chi(G)$ colours; they are called {\em ugly graphs}. We show that no perfect graph is ugly. We also give simple proofs of this fact for subclasses of perfect graphs (block graphs, comparability graphs), and show that no $K_4$-minor free graph is ugly. Moreover, our proofs are constructive, and imply the existence of polynomial-time algorithms to compute good connected orderings for these graph classes.

When D-companion matrix meets incomplete polynomials

In this paper, we provide a simple proof of a generalization of the Gauss-Lucas theorem. By using methods of D-companion matrix, we obtain a majorization relationship between the zeros of convex combinations of incomplete polynomials and the original polynomial. Moreover, we prove that the set of zeros of all convex combinations of incomplete polynomials coincides with the closed convex hull of zeros of the original polynomial. Location of zeros of convex combinations of incomplete polynomials is determined.

On Rotfel'd type inequalities for sectorial matrices

Using a matrix decomposition for sectorial matrices and an AM-GM inequality for singular values, we present a new Rotfel'd type inequality for sectorial matrices, which generalizes and partially improves some results by Zhang. We also give alternative simpler proofs of a main result of a recent paper by Yang, Lu and Chen as well as that by Fu and Liu.

A simple proof of the Wiener–Ikehara Tauberian Theorem

The Wiener–Ikehara Tauberian theorem is an important theorem giving an asymptotic formula for the sum of coefficients of a Dirichlet series ∑n=1∞a(n)ns. We provide a simple and elegant proof of the Wiener–Ikehara Tauberian theorem which relies only on basic Fourier analysis and known estimates for the given Dirichlet series. This method also allows us to derive a version of the Wiener–Ikehara theorem with an error term.

Improved quantum data analysis

We provide more sample-efficient versions of some basic routines in quantum data analysis, along with simpler proofs. Particularly, we give a quantum "Threshold Search" algorithm that requires only $O((\log^2 m)/\epsilon^2)$ samples of a $d$-dimensional state $\rho$. That is, given observables $0 \le A_1, A_2, ..., A_m \le 1$ such that $\mathrm{tr}(\rho A_i) \ge 1/2$ for at least one $i$, the algorithm finds $j$ with $\mathrm{tr}(\rho A_j) \ge 1/2-\epsilon$. As a consequence, we obtain a Shadow Tomography algorithm requiring only $\tilde{O}((\log^2 m)(\log d)/\epsilon^4)$ samples, which simultaneously achieves the best known dependence on each parameter $m$, $d$, $\epsilon$. This yields the same sample complexity for quantum Hypothesis Selection among $m$ states; we also give an alternative Hypothesis Selection method using $\tilde{O}((\log^3 m)/\epsilon^2)$ samples.

How Schwarzschild could have discovered and fixed the problem with his metric

Karl Schwarzschild's landmark paper revealed the first ever exact solution to Einstein's gravitational field equation. This was a major scientific achievement. However, his solution for the metric of a spherically symmetric spacetime is different than the one found in textbooks today—because Schwarzschild assumed, like in Newtonian gravity, that only one singularity could exist. Given the newness of Einstein's general theory of relativity and the existing paradigm of Newtonian gravity, his assumption was natural. While other authors have previously pointed out this shortcoming and corrected it, the contribution of this paper is to show how Schwarzschild might have discovered it himself. A simple geometric proof indicates how Schwarzschild, with a different assignment of a single constant of integration, would have arrived at the metric found in textbooks today.

A nonlocal Lagrangian traffic flow model and the zero-filter limit

In this study, we start from a Follow-the-Leaders model for traffic flow that is based on a weighted harmonic mean (in Lagrangian coordinates) of the downstream car density. This results in a nonlocal Lagrangian partial differential equation (PDE) model for traffic flow. We demonstrate the well-posedness of the Lagrangian model in the L1\\documentclass[12pt]{minimal} \\usepackage{amsmath} \\usepackage{wasysym} \\usepackage{amsfonts} \\usepackage{amssymb} \\usepackage{amsbsy} \\usepackage{mathrsfs} \\usepackage{upgreek} \\setlength{\\oddsidemargin}{-69pt} \\begin{document}$$L^1$$\\end{document} sense. Additionally, we rigorously show that our model coincides with the Lagrangian formulation of the local LWR model in the “zero-filter” (nonlocal-to-local) limit. We present numerical simulations of the new model. One significant advantage of the proposed model is that it allows for simple proofs of (i) estimates that do not depend on the “filter size” and (ii) the dissipation of an arbitrary convex entropy.

Linkage and F-Regularity of Determinantal Rings

Abstract In this paper, we prove that the generic link of a generic determinantal ring defined by maximal minors is strongly $F$-regular. In the process, we strengthen a result of Chardin and Ulrich in the graded setting. They showed that the generic residual intersections of a complete intersection ring with rational singularities again have rational singularities. We show that if the said complete intersection is defined by homogeneous elements and is $F$-rational, then in fact, its generic residual intersections are strongly $F$-regular in positive prime characteristic. Hochster and Huneke showed that determinantal rings are strongly $F$-regular; however, their proof is quite involved. Our techniques allow us to give a new and simple proof of the strong $F$-regularity of determinantal rings defined by maximal minors.