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Let $\mathcal{F}$ be a family of subsets of $[n]$. The diameter of $\mathcal{F}$ is the maximum size of symmetric differences among pairs of its members. Resolving a conjecture of Erdős, Kleitman determined the maximum size of a family with fixed diameter, which states that a family with diameter $s$ has cardinality at most that of a Hamming ball of radius $s/2$. Specifically, if $\mathcal{F} \subseteq 2^{[n]}$ is a family with diameter $s$, then for $s=2d$, $|\mathcal{F}|\le \sum_{i=0}^d {n \choose i}$; for $s=2d+1$, $|\mathcal{F}|\le \sum_{i=0}^d {n \choose i} + {n-1 \choose d}$. This result is known as the Kleitman diameter theorem, which generalizes both the Katona union theorem and the Erdős--Ko--Rado theorem. In 2017, Frankl provided a complete characterization of the extremal families of Kleitman's theorem and provided a stability result. In this paper, we determine the extremal families of Frankl's theorem and establish a further stability result of Kleitman's theorem. This solves a recent problem proposed by Li and Wu. Our findings constitute the second stability for the Kleitman diameter theorem.

We discuss an incompleteness result proven by Bezboruah and Shepherdson. This result tells us that the weak theory ${\sf PA}^-$ does not prove the consistency of any theory (under certain assumptions explained in the paper). Kreisel argued that such a result is not meaningful. We discuss Kreisel's objection and conclude that his argument does not hold water. We compare Pudlák's extension of the Second Incompleteness Theorem with the Bezboruah-Sheperdson Theorem. Finally, we reprove the Bezboruah-Sheperdson Theorem for a sequence coding based on an insight of Nielsen and Markov.

The goal of this article is to derive the reciprocity theorem, mutual energy theorem from Poynting theorem instead of from Maxwell equation. The Poynting theorem is generalized to the modified Poynting theorem. In the modified Poynting theorem the electromagnetic field is superimposition of different electromagnetic fields including the retarded potential and advanced potential, time-offset field. The media epsilon (permittivity) and mu (permeability) can also be different in the different fields. The concept of mutual energy is introduced which is the difference between the total energy and self-energy. Mixed mutual energy theorem is derived. We derive the mutual energy from Fourier domain. We obtain the time-reversed mutual energy theorem and the mutual energy theorem. Then we derive the mutual energy theorem in time-domain. The instantaneous modified mutual energy theorem is derived. Applying time-offset transform and time integral to the instantaneous modified mutual energy theorem, the time-correlation modified mutual energy theorem is obtained. Assume there are two electromagnetic fields one is retarded potential and one is advanced potential, the convolution reciprocity theo

A weak version of Birkhoff's generalization of the Perron-Frobenius theorem states that every endomorphism of a finite-dimensional real vector that leaves invariant a non-degenerate closed convex cone has an eigenvector in that cone. Here, we show that this theorem, whose proof relies only upon basic convex analysis, yields very short proofs of both the spectral theorem for selfadjoint operators of Euclidean spaces and the Fundamental Theorem of Algebra.

Two celebrated extensions of Helly's theorem are the Fractional Helly theorem of Katchalski and Liu (1979) and the Quantitative Volume theorem of Bárány, Katchalski, and Pach (1982). Improving on several recent works, we prove an optimal combination of these two results. We show that given a family $\mathcal{F}$ of $n$ convex sets in $\mathbb{R}^d$ such that at least $α\binom{n}{d+1}$ of the $(d+1)$-tuples of $\mathcal{F}$ have an intersection of volume at least 1, then one can select $Ω_{d,α}(n)$ members of $\mathcal{F}$ whose intersection has volume at least $Ω_d(1)$. Furthermore, with the help of this theorem, we establish a quantitative version of the $(p,q)$ theorem of Alon and Kleitman. Let $p\geq q\geq d+1$ and let $\mathcal{F}$ be a finite family of convex sets in $\mathbb{R}^d$ such that among any $p$ elements of $\mathcal{F}$, there are $q$ that have an intersection of volume at least $1$. Then, we prove that there exists a family $T$ of $O_{p,q}(1)$ ellipsoids of volume $Ω_d(1)$ such that every member of $\mathcal{F}$ contains at least one element of $T$. Finally, we present extensions about the diameter version of the Quantitative Helly theoerm.

A classical result of Sidorenko (1989) shows that the Turán density of every $r$-uniform hypergraph with three edges is bounded from above by $1/2$. For even $r$, this bound is tight, as demonstrated by Mantel's theorem on triangles and Frankl's theorem on expanded triangles. In this note, we prove that for odd $r$, the bound $1/2$ is never attained, thereby answering a question of Keevash and revealing a fundamental difference between hypergraphs of odd and even uniformity. Moreover, our result implies that the expanded triangles form the unique class of three-edge hypergraphs whose Turán density attains $1/2$.

A new theorem for black holes is found. It is called the horizon mass theorem. The horizon mass is the mass which cannot escape from the horizon of a black hole. For all black holes: neutral, charged or rotating, the horizon mass is always twice the irreducible mass observed at infinity. Previous theorems on black holes are: 1. the singularity theorem, 2. the area theorem, 3. the uniqueness theorem, 4. the positive energy theorem. The horizon mass theorem is possibly the last general theorem for classical black holes. It is crucial for understanding Hawking radiation and for investigating processes occurring near the horizon.

In this short note, we will explain that the good moduli space morphisms behave as if they are proper when we consider sheaf operations, though they are not separated. For example, the decomposition theorem and the base change theorem hold for these morphisms, which have applications to the cohomological study of moduli spaces.

When the Canonical Ramsey's Theorem by Erdős and Rado is applied to regressive functions one obtains the Regressive Ramsey's Theorem by Kanamori and McAloon. Taylor proved a "canonical" version of Hindman's Theorem, analogous to the Canonical Ramsey's Theorem. We introduce the restriction of Taylor's Canonical Hindman's Theorem to a subclass of the regressive functions, the $λ$-regressive functions, relative to an adequate version of min-homogeneity and prove some results about the Reverse Mathematics of this Regressive Hindman's Theorem and of natural restrictions of it. In particular we prove that the first non-trivial restriction of the principle is equivalent to Arithmetical Comprehension. We furthermore prove that this same principle strongly computably reduces the well-ordering-preservation principle for base-$ω$ exponentiation.

Krieger's embedding theorem provides necessary and sufficient conditions for an arbitrary subshift to embed in a given topologically mixing $\mathbb{Z}$-subshift of finite type. For some $\mathbb{Z}^d$-subshifts of finite type, Lightwood characterized the \emph{aperiodic} subsystems. In the current paper we prove a new embedding theorem for a class of subshifts of finite type over any countable abelian group. Our main theorem provides necessary and sufficient conditions for an arbitrary subshift $X$ to embed inside a given subshift of finite type $Y$ that satisfies a certain condition. For the particular case of $\mathbb{Z}$-subshifts, our new theorem coincides with Krieger's theorem. In particular, our result gives the first complete characterization of the subsystems of the multidimensional full shift $Y= A^{\mathbb{Z}^d}$. The natural condition on the target subshift $Y$, introduced explicitly for the first time in the current paper, is called the map extension property. It was introduced implicitly by Mike Boyle in the early 1980's for $\mathbb{Z}$-subshifts, and is closely related to the notion of an absolute retract, introduced by Borsuk in the 1930's. A $\mathbb{Z}$-subshift

Kurzweil's theorem ('55) is concerned with zero-one laws for well approximable targets in inhomogeneous Diophantine approximation under the badly approximable assumption. In this article, we prove the divergent part of a Kurzweil type theorem via a suitable construction of ubiquitous systems when the badly approximable assumption is relaxed. Moreover, we also discuss some counterparts of Kurzweil's theorem.

The aim of this survey papier is to present a result due to Eisenstein, to prove a generalized version of it, and to present some applications of this Eisenstein's Theorem, in particular to the study of the algebraic closure of the field of power series in several indeterminates.

In this paper, we give a form of refined Roth's theorem. As an application, we prove a special case of the $abc$-conjecture.

We establish new versions of the Wiener-Ikehara theorem where only boundary assumptions on the real part of the Laplace transform are imposed. Our results generalize and improve a recent theorem of T. Koga [J. Fourier Anal. Appl. 27 (2021), Article No. 18]. As an application, we give a quick Tauberian proof of Blackwell's renewal theorem.

Contextuality is a key feature of quantum mechanics, as was first brought to light by Bohr and later realised more technically by Kochen and Specker. Isham and Butterfield put contextuality at the heart of their topos-based formalism and gave a reformulation of the Kochen-Specker theorem in the language of presheaves. Here, we broaden this perspective considerably (partly drawing on existing, but scattered results) and show that apart from the Kochen-Specker theorem, also Wigner's theorem, Gleason's theorem, and Bell's theorem relate fundamentally to contextuality. We provide reformulations of the theorems using the language of presheaves over contexts and give general versions valid for von Neumann algebras. This shows that a very substantial part of the structure of quantum theory is encoded by contextuality.

In this paper we present a series of seemingly unrelated results of Complex Analysis which are in fact connected via a different approach to their proofs using the results of Errett Bishop of volumes and limits of analytic varieties. We start by proving Chow's theorem by a technique suggested long time ago in the beautiful book by Gabriel Stolzenberg. We think this approach is very attractive and easier for students and newcomers to understand; also the theory presented here is linked to areas of mathematics that are not usually associated with Chow's result. In addition, Bishop's results imply both Chow's and Remmert-Stein's theorems directly, meaning that this view is simpler and just as profound as Remmert-Stein's proof. After that, we give a comparison table that explains how Bishop's theorems generalize to several complex variables classical results of one complex variable and prove Montel's compactness theorem using the techniques presented here. Finally we give an alternative proof of a theorem of Edwards, Millet and Sullivan of foliations with compact leaves for the case of complex foliations in Kähler manifolds.

We give a new proof, along with some generalizations, of a folklore theorem (attributed to Laurent Lafforgue) that a rigid matroid (i.e., a matroid with indecomposable basis polytope) has only finitely many projective equivalence classes of representations over any given field.

The Tait-Kneser theorem, first demonstrated by Peter G. Tait in 1896, states that the osculating circles along a plane curve with monotone non-vanishing curvature are pairwise disjoint and nested. This note contains a proof of this theorem using the Lorentzian geometry of the space of circles. We show how a similar proof applies to two variations on the theorem, concerning the osculating Hooke and Kepler conics along a plane curve. We also prove a version of the 4-vertex theorem for Kepler conics.

A theorem which is named after the American Mathematician Moris Marden states a very surprising and interesting fact concerning the relationship between the points of a triangle in the complex plane and the zeros of two complex polynomials related to this triangle: "Suppose the zeroes z1, z2, and z3 of a third-degree polynomial p(z) are non-collinear. There is a unique ellipse inscribed in the triangle with vertices z1, z2, z3 and tangent to the sides at their midpoints: the Steiner in-ellipse. The foci of that ellipse are the zeroes of the derivative p'(z)." (Wikipedia contributors, "Marden's theorem", http://en.wikipedia.org/wiki/Marden%27s_theorem). This document describes how Scilab, a popular and powerful open source alternative to MATLAB, can be used to visualize the above stated theorem for arbitrary complex numbers z1, z2, and z3 which are not collinear. It is further demonstrated how the equations of the Steiner ellipses of a triangle in the complex plane can be calculated and plotted by applying this theorem.