Extremal graphs with bounded matching number: a short proof of the Alon-Frankl theorem

The notion of essential submodules and essential extensions of modules are extended to groups (typically nonabelian), and several necessary and sufficient conditions for a group to possess a proper essential subgroup are investigated. Further, we have completely characterized groups that do not possess a proper essential extension. These observations are used in concluding several properties of groups having essential subgroups. Finally, a short proof of the well-known theorem of Eilenberg and Moore that the only injective object in the category of groups is the trivial group is given.

Using the Mehelt-Fock-Clifford transform, we define generalized modulus of smoothness in the space \(L^2(J;x^{-\frac{1}{2}}dx)\). Based on the kernel \(P_{i\sqrt{\lambda}-\frac{1}{2}}\) and the operator \(A_x^m\) we define Sobolev-type and \(K\)-functionals. The main result of the paper is the proof of the equivalence theorem for a \(K\)-functional and a modulus of smoothness for the Mehler-Fock-Clifford transform.

This note focuses on alternative proofs of Darboux's theorem for differentiable functions and Bolzano's theorem for continuous functions. These results are derived from each other using the Lagrange mean value theorem and the fundamental theorem of integral calculus.

We correct an error in the proof of Theorem 11 in our paper “Approximate Degradable Quantum Channels”, IEEE Trans. Inf. Theory, vol. 63, no. 12, pp. 7832-7844, 2017, concerning an upper bound on the private capacity of an approximate anti-degradable channel. Furthermore, we show how to obtain a tighter bound for the quantum capacity.

We generalize a well-known result proved by Filaseta and Trifonov in 2002 that the Bessel polynomials of all degrees are irreducible over the field of rational numbers. The proof given here of our generalization appears to be simpler than the known proofs of Filaseta-Trifonov Theorem. We use some recent results by Lehmer, Luca, Najman and Shorey regarding the largest prime divisor of a product of consecutive integers, which play a significant role in making the proof shorter. The backbone of our proof is a theorem based on an extension introduced by Ore of the concept of Newton polygons. We illustrate the utility of our generalization by showing that it leads to new classes of monic irreducible polynomials with integer coefficients for which the known irreducibility criteria for polynomials do not seem to be applicable.

Abstract This paper discusses an alternative to the classical arguments used to obtain the localized laws of momentum balance, i.e. Cauchy's stress theorem and the equations of motion. We show they can be obtained from the integral form of momentum balance using only the localization theorem without any asymptotic analysis of small parts. A key step in the process is to recast the integral momentum balance equation into an equivalent form before performing localization. This modified form leads to a direct deduction of the equations of motion using the localization theorem. Cauchy's stress theorem follows shortly afterward. Since the other localized balance laws of thermomechanics are commonly found via the localization theorem, it is hoped that the simple derivation provided herein could serve a pedagogical purpose, providing for a unified argument structure by which all standard balance laws in classical mechanics can be deduced. The derivation can also be generalized to produce an alternate proof of Noll's theorem, which shows that the traction on a surface at a point depends only on the outward normal of the surface and not any of its higher-order local descriptors such as curvature, also known as Cauchy's postulate.

We prove an r -variation estimate, r > 4, in the norm for ergodic averages with respect to three commuting transformations.It is not known whether such estimates hold for all r 2 as in the analogous cases for one or two commuting transformations, or whether such estimates hold for any r < for more than three commuting transformations.1. Introduction 539 2. A collection of propositions on singular Brascamp-Lieb forms 545 3. Proof of Theorem 1.2 from Proposition 2.1 and Corollaries 2.2 and 2.4 553 4. Proof of Proposition 2.1 using Propositions 2.3 and 2.5 559 5. Proof of Proposition 2.3 using Propositions 2.5 and 2.6 561 6. Proof of Proposition 2.5 using Lemma 3 in [Durcik and Thiele 2020] 563 7. Proof of Proposition 2.6 using Propositions 2.7 and 2.8 569 8. Proof of Proposition 2.7 using Propositions 2.8, 2.9, and [Durcik et al. 2022] 576 9. Proof of Propositions 2.8 and 2.9

It is well known that a numerical scheme for solving a hyperbolic conservation law, when convergent, may not always converge to a weak solution. The remedy is the famous Lax-Wendroff theorem, stating that a conservative numerical scheme, when convergent, always converges to a weak solution of the conservation law. In this paper, we address the same issue for numerically solving degenerate non-linear diffusion or convection-diffusion equations. We start with an example showing that a conservative scheme, in the sense of that for conservation laws, may still converge to a function which is not a weak solution of the degenerate non-linear diffusion equation. We then introduce a stronger form of conservative schemes, which we term as doubly-conservative (DoC) schemes, that would allow the proof of a Lax-Wendroff type theorem, namely if a DoC scheme converges, then it will converge to the weak solution of the degenerate non-linear diffusion or convection-diffusion equation. The concept of DoC schemes is applicable to high order schemes, on one-dimensional non-uniform meshes as well as on two-dimensional Cartesian meshes and unstructured triangular meshes. Finally, we design a new DoC local discontinuous Galerkin scheme that remains semi-discrete stable even when the diffusion coefficient degenerates. Numerical experiments demonstrate optimal convergence rates of this new scheme and validate our theoretical results.

We present an elementary proof of the adiabatic theorem for Markov jump processes. This theorem, analogous to its quantum counterpart, states that a system starting from an instantaneous steady state remains in the instantaneous steady state if the transition rate matrix changes sufficiently slowly. Our approach adapts techniques from quantum adiabatic theorems while addressing the challenges posed by non-Hermitian dynamics. We highlight the unique properties of transition matrices that enable this proof, providing a more accessible foundation for understanding adiabatic behavior in Markov jump processes.

In this paper, we present an alternative proof of the Coisotropic Embedding Theorem in which the geometric choice of a connection is recast as the algebraic choice of an embedding into the cotangent bundle. The symplectic thickening is then identified as the submanifold determined by the Hamiltonian momenta conjugate to the kernel directions of the pre-symplectic form.

Revised proofs of Kenneth Arrow's impossibility theorem, one of the most influential theorems in economics, political science, and philosophy, have been presented in prose form, incorporating novel ideas such as decisive sets and pivotal voters. This study develops another approach to proving the theorem. Using a proof calculus in formal logic, we construct a proof with a full mathematical representation. While previous proofs emphasize intuitive accessibility, this one focuses on meticulous derivation and reveals the global structure of the social welfare function central to the theorem. The primary aim is to contribute methodologically to research on the theorem by demonstrating the effectiveness of systematically applying techniques from formal logic to its proof. Additionally, it accommodates a broader range of preference relations than those typically considered rational in standard economic models, allowing for the integration of diverse human behavior patterns into a single theoretical framework. The interdisciplinary relevance of the theorem is also discussed, including its relation to epistemology and philosophy.

According to general relativity, black holes are incomplete, which prevents developing a complete physical description of their dynamical formation and evolution once quantum effects are taken into account. Theories beyond general relativity may provide a more complete description of black hole interiors. In this work, the most general form of the field equations for spherically symmetric gravitational fields, in which the Einstein tensor is deformed into a conserved tensor constructed from up to second-order derivatives of the metric, is described. These equations set up the stage for the study of the dynamics of spherically symmetric spacetimes beyond general relativity, providing tools for the theoretical exploration of a paradigm of black hole physics free of the incompleteness characteristic of Einstein's theory. A general proof of the Birkhoff-Jebsen theorem for vacuum solutions, and the construction of field equations describing the effective geometrodynamics of regular black holesinteracting with matter, are discussed.

Error correction is an essential part of the theory of quantum computation. However, new quantum computation students may find the theories of error correction and fault tolerance daunting, or they may be stuck with theoretical/outdated schemes (such as the one in the original proof of the threshold theorem by Aharonov and Ben-or) with unrealistically low thresholds and/or high overhead. In this article, we describe an adequately modern approach to fault-tolerant quantum computation based on the surface code and lattice surgery. The reader is assumed to have a basic understanding of quantum computation (state vectors, unitary gates, and measurements, etc.), but no prior knowledge about quantum codes or quantum error correction is needed.

The claim and the proof of Theorem 2.9 in Vassilev [Trans. Amer. Math. Soc. 363 (2011), no. 1, 37–62] are not correct as stated. We give a correction in Section 1.6. In addition, we correct several typos in the text and strengthen slightly some results of Vassilev [Trans. Amer. Math. Soc. 363 (2011), no. 1, 37–62].

In this work, we propose a novel convolution product associated with the H -transform, denoted by ∗ H , and explore its fundamental properties. Here, the H -transform may be regarded as a refined variant of the classical Fourier, Hartley transform, with kernel function depending on two parameters a, b. Our first contribution shows that the space of integrable functions, equipped with multiplication given by the ∗ H -convolution, constitutes the commutative Banach algebra over the complex field, albeit without an identity element. Second, establishes the Wiener–Lévy type invertibility criterion for H -algebras, obtained through the density property and process of unitarization, which serves as a key step toward the proof of Gelfand's spectral radius theorem. Third, provides an explicit upper-bound of Young's inequality for ∗ H -convolution and its direct corollary. Finally, all of these theoretical findings are applied to analyse specific classes of the Fredholm integral equations and heat source problems, yielding a priori estimates under the established assumptions.

A simple proof of a reverse Minkowski theorem for integral lattices

In this article we present three different proofs of the spectral theorem. Compared to the standard proofs using determinants or Lagrange multipliers, the three proofs presented have their own pedagogical advantages and deserve to reach a broader audience.

In this paper, we consider certain special sine, tangent, and cotangent symmetric matrices, along with their associated integer matrices. We present interesting relationships that connect these matrices. Relevant examples of the corresponding matrices are also provided, along with several intriguing problems. A separate section includes proofs of the main theorems, which we believe are noteworthy in their own right.

Another proof of the fundamental theorem of arithmetic?