Alternative Proof Research Articles

On Strongly Regular Graphs and the Friendship Theorem

This paper presents an alternative proof of the celebrated friendship theorem, originally established by Erdős, Rényi, and Sós in 1966. The proof relies on a closed-form expression for the Lovász ϑ-function of strongly regular graphs, recently derived by the author. Additionally, this paper considers some known extensions of the theorem, offering discussions that provide insights into the friendship theorem, one of its extensions, and the proposed proof. Leveraging the closed-form expression for the Lovász ϑ-function of strongly regular graphs, this paper further establishes new necessary conditions for a strongly regular graph to be a spanning or induced subgraph of another strongly regular graph. In the case of induced subgraphs, the analysis also incorporates a property of graph energies. Some of these results are extended to regular graphs and their subgraphs.

Remarks on second and third weights of projective Reed–Muller codes

Determining the weight distributions of the projective Reed–Muller codes is a very hard problem and has been studied extensively in the literature. In this paper, we provide an alternative proof of the second weight of the projective Reed–Muller codes [Formula: see text] where [Formula: see text] and [Formula: see text]. We show that the second weight is attained by codewords that correspond to hypersurfaces containing a hyperplane under the hypothesis on [Formula: see text]. Furthermore, we compute the second weight of [Formula: see text] for [Formula: see text]. Furthermore, we give an upper bound for the third weight of [Formula: see text].

Inequality-free proof of Bell's theorem

Bell's theorem supposedly demonstrates an irreconcilable conflict between quantum mechanics and local realistic hidden-variable theories. Most proofs of Bell's theorem, are based on inequalities. In this paper we present an alternative proof which does not involve inequalities, but only a direct comparison between correlation functions calculated using quantum mechanics on the one hand, and those calculated according to local realistic hidden-variable theories on the other. Our proof is based on a physically motivated use of Fourier series for periodic functions, and confirms that local realistic hidden-variable theories are incompatible with quantum mechanics. Published by the American Physical Society 2025

Asymptotic error of Bonferroni procedure under weak dependence via Chen–Stein method

Abstract In this paper, we are interested in the limiting distribution of the number of false positives when using Bonferroni adjustment for large-scale multiple testing, as the number of hypotheses grows to infinity. It is proven in the literature that the distribution converges to a Poisson distribution, if the statistics are positively equi-correlated normal but nearly independent. In this paper, we provide an alternative proof using the Chen–Stein method. Unlike existing works, our proof provides a rate of convergence of the number of false positives to its asymptotic distribution, which we also confirm numerically. In addition, we show that our results are applicable to a more generalized setting beyond the equi-correlation assumption.

Alternative proofs of the GNKD and KKMS lemmas: a game-theoretic underpinning

In this paper, we provide alternative proofs of fundamental results of Walrasian general equilibrium theory, and of cooperative game theory, by viewing them as results pertaining to a two-person qualitative or a one-person generalized game: in short, by giving them a game-theoretic underpinning. Under the first category, we have the Gale–Nikaido–Kuhn–Debreu (GNKD) lemma in mind, and under the second, the Knaster–Kuratowski–Mazurkiewicz–Shapley (KKMS) lemma.

On the Orbital Stability of Gross-Pitaevskii Solitons

The one-dimensional Gross-Pitaevskii equation, under non-vanishing boundary condition, has a set of solitary solutions. The orbital stability of these solitons has been well established. However, the existing proof methods usually treat the cases of dark solitons and black solitons separately. Here we provide an alternative proof of this orbital stability result, which treats the two cases in a unified framework.

Extensions of Schoen–Simon–Yau and Schoen–Simon theorems via iteration à la De Giorgi

We give an alternative proof of the Schoen–Simon–Yau curvature estimates and associated Bernstein-type theorems (Schoen et al. in Acta Math. 134:275–288, 1975), and extend the original result by including the case of 6-dimensional (stable minimal) immersions. The key step is an ε-regularity theorem, that assumes smallness of the scale-invariant L2 norm of the second fundamental form. Further, we obtain a graph description, in the Lipschitz multi-valued sense, for any stable minimal immersion of dimension n≥2, that may have a singular set Σ of locally finite Hn−2-measure, and that is weakly close to a hyperplane. (In fact, if the Hn−2-measure of the singular set vanishes, the conclusion is strengthened to a union of smooth graphs.) This follows directly from an ε-regularity theorem, that assumes smallness of the scale-invariant L2 tilt-excess (verified when the hypersurface is weakly close to a hyperplane). Specialising the multi-valued decomposition to the case of embeddings, we recover the Schoen–Simon theorem (Schoen and Simon 34:741–797, 1981). In both ε-regularity theorems the relevant quantity (respectively, length of the second fundamental form and tilt function) solves a non-linear PDE on the immersed minimal hypersurface. The proof is carried out intrinsically (without linearising the PDE) by implementing an iteration method à la De Giorgi (from the linear De Giorgi–Nash–Moser theory). Stability implies estimates (intrinsic weak Caccioppoli inequalities) that make the iteration effective despite the non-linear framework. (In both ε-regularity theorems the method gives explicit constants that quantify the required smallness.)

On Torelli groups and Dehn twists of smooth 4‐manifolds

AbstractThis note has two related but independent parts. Firstly, we prove a generalisation of a recent result of Gay on the smooth mapping class group of . Secondly, we give an alternative proof of a consequence of work of Saeki, namely that the Dehn twist along the boundary sphere of a simply connected closed smooth 4‐manifold with is trivial after taking connected sums with enough copies of .

Non-stationary Itô–Kawada and ergodic theorems for random isometries

We consider a non-stationary sequence of independent random isometries of a compact metrizable space. Assuming that there are no proper closed subsets with deterministic image, we establish a weak-* convergence to the unique invariant under isometries measure, ergodic theorem and large deviation type estimate. We also show that all the results can be carried over to the case of a random walk on a compact metrizable group. In particular, we prove a non-stationary analog of classical Itô–Kawada theorem and give a new alternative proof for the stationary case.

Study of the cone of sums of squares plus sums of nonnegative circuit forms

Abstract In this article, we combine sums of squares (SOS) and sums of nonnegative circuit (SONC) forms, two independent nonnegativity certificates for real homogeneous polynomials. We consider the convex cone SOS+SONC of forms that decompose into a sum of an SOS and a SONC form and study it from a geometric point of view. We show that the SOS+SONC cone is proper and neither closed under multiplication nor under linear transformation of variables. Moreover, we present an alternative proof of an analog of Hilbert’s 1888 Theorem for the SOS+SONC cone and prove that in the non-Hilbert cases it provides a proper superset of the union of the SOS and SONC cones. This follows by exploiting a new necessary condition for membership in the SONC cone.

FORMAL LIFTING OF DUALIZING COMPLEXES AND CONSEQUENCES

Abstract We show that for a Noetherian ring A that is I-adically complete for an ideal I, if $A/I$ admits a dualizing complex, so does A. This gives an alternative proof of the fact that a Noetherian complete local ring admits a dualizing complex. We discuss several consequences of this result. We also consider a generalization of the notion of dualizing complexes to infinite-dimensional rings and prove the results in this generality. In addition, we give an alternative proof of the fact that every excellent Henselian local ring admits a dualizing complex, using ultrapower.

A note on von Neumann regular matrices

We give a new perspective on some recent results of Chiru and Crivei on von Neumann regular matrices, including an alternative matrix proof of a characterization of von Neumann regular matrices of Northcott's. We present several conditions on a ring R under which every matrix over R whose determinantal and McCoy ranks coincide is von Neumann regular.

Refinements of Generic Algebraic Inequalities with Application

In this paper, we give alternative proofs and refinements of two generic inequalities due to Bagul and Dhaigude in 2022. As a result, Huygens-type inequalities are established. Illustrations are given to support the main results.

Attempts and Inferences of the Four-Color Theorem

The Four-Color Theorem is a classic problem in graph theory, stating that any planar map can be colored using no more than four colors so that no adjacent regions share the same color. Since 1976, when Appel and Haken used computer assistance to prove this theorem, it has been considered solved. However, due to its complexity and the difficulty of manually verifying the proof, some mathematicians still have doubts. This study proposes a new logical approach to provide an alternative proof for the Four-Color Theorem. Using a combination of theoretical derivations and graph theory tools, the paper first analyzes the basic structures of planar graphs, then use inductive reasoning to confirm coloring patterns in small graphs, gradually extending the method to more complex graphs. It developed a specific algorithm that simplifies vertices and edges in planar graphs, showing that any complex planar graph can be reduced to a few basic shapes, allowing it to be colored with fewer than four colors. Our results show that this reasoning model can confirm the Four-Color Theorem for a range of planar graphs without computer assistance. Though not covering all graph types, this model offers new insights for future research, especially in simplifying large graphs.

EXTENDING THE IDEALS OF SEMIRINGS TO FUZZY SETTING BY METATHEOREM

Notions of different types of fuzzy ideals are investigated in a semiring in the light of Tom Head’s metatheorem. Using Rep function, the classes of these fuzzy ideals are shown to be projection closed. The metatheorem was formulated to obtain fuzzy versions of classical outcomes. Tom Head’s metatheorem is also employed to present alternative proofs for various results related to fuzzy ideals, offering a non-computational approach. The proofs obtained by metatheorem are simple, shorter and calculation free. The metatheorem is utilized to characterize a semiring that is regular, intra-regular and weakly regular in terms of various types of fuzzy ideals.

A NOTE ON THE LARGE SIEVE INEQUALITY FOR MODULI GENERATED BY A QUADRATIC

Abstract We develop a generalisation of the square sieve of Heath-Brown and use it to give an alternate proof of one of the large sieve inequalities in our previous paper [‘A large sieve inequality for characters to quadratic moduli’, Preprint, https://web.maths.unsw.edu.au/~ccorrigan/preprint6.pdf].

Apartment classes of integral symplectic groups

In this paper, we present an alternative proof of a theorem of Gunnells, which states that the Steinberg module of [Formula: see text] is a cyclic [Formula: see text]-module, generated by integral apartment classes.

A Proof Using Böhme's Lemma That no Petersen Family Graph has a Flat Embedding

Sachs and Conway-Gordon used linking number and a beautiful counting argument to prove that every graph in the Petersen family is intrinsically linked (have a pair of disjoint cycles that form a nonsplit link in every spatial embedding) and thus each family member has no flat spatial embedding (an embedding for which every cycle bounds a disk with interior disjoint from the graph). We give an alternate proof that every Petersen family graph has no flat embedding by applying Böhme's Lemma and the Jordan-Brouwer Separation Theorem.

The hybrid structures for two-dimensional Deligne-Mostow lattices

The hybrid structures for two-dimensional Deligne-Mostow lattices

Herr complex of (φ,τ)-modules

Herr complex of (φ,τ)-modules