Abstract We reconsider velocity addition/subtraction in Special Relativity (SR) and re-derive its well-known non-commutative and non-associative algebraic properties in a self contained way, including various explicit expressions for the Thomas angle, the derivation of which will be seen to be not as challenging as often suggested. All this is based on the polar-decomposition theorem in the traditional component language, in which Lorentz transformations are ordinary matrices. In the second part of this paper we offer a less familiar alternative geometric view, that leads to an invariant definition of the concept of relative velocity between two states of motion, which is based on the boost-link-theorem, of which we also offer an elementary proof that does not seem to be widely known in the relativity literature. Finally we compare this to the corresponding geometric definitions in Galilei-Newton spacetime, emphasising similarities and differences. Regarding the presentation of the material we will pursue an uncompromising pedagogical strategy, willingly accepting repetitions and occasional redundancies if deemed beneficial for clarity and the avoidance of anticipated misunderstandings. An appendix with four sections includes some mathematical details on results needed in the main text, as well some recollections on notions like semi-direct products of groups and affine spaces.

An elementary proof of the beta-gamma function identity is presented by Chen and Chen [Chen, H.-C., & Chen, H. L. (2025). The beta-gamma function identity via Taylor's formula. International Journal of Mathematical Education in Science and Technology. https://doi.org/10.1080/0020739X.2025.2573176]. In this note we remark that the limiting arguments in Section 2.3 and Appendix of Chen and Chen [Chen, H.-C., & Chen, H. L. (2025). The beta-gamma function identity via Taylor's formula. International Journal of Mathematical Education in Science and Technology. https://doi.org/10.1080/0020739X.2025.2573176] can be considerably simplified. Our alternative arguments make their proof much more accessible to students in elementary single-variable calculus classes.

Neural network quantum states emerge as a promising tool for solving quantum many-body problems. However, its successes and limitations are still not well-understood, in particular for fermions with complex signstructures. Based on our recent work [Z. Wu et al., J. Chem. Theory Comput. 21, 10252-10262 (2025)], we generalize the restricted Boltzmann machine Ansatz to a more general class of states for fermions, which is formed by the product of neurons and, hence, will be referred to as neuron product states (NPS). NPS builds correlation in a very different way compared with the closely related correlator product states [H. J. Changlani et al., Phys. Rev. B 80, 245116 (2009)], which use full-rank local correlators. In contrast, each correlator in NPS contains long-range correlations across all the sites, with its representational power constrained by the simple function form. We prove that products of such simple nonlocal correlators can approximate any wavefunction arbitrarily well under certain mild conditions on the form of activation functions. In addition, we also provide elementary proofs for the universal approximation capabilities of feedforward neural networks and neural network backflow in second quantization. Together, these results provide a deeper insight into the neural network representation of many-body wavefunctions in second quantization.

Abstract Calderón’s inverse conductivity problem has, so far, only been subject to conditional logarithmic stability for infinite-dimensional classes of conductivities and to Lipschitz stability when restricted to finite-dimensional classes. Focusing our attention on the unit ball domain in any spatial dimension $$d\ge 2$$ d ≥ 2 , we give an elementary proof that there are (infinitely many) infinite-dimensional classes of conductivities for which there is Lipschitz stability. In particular, Lipschitz stability holds for general expansions of conductivities, allowing all angular frequencies but with limited freedom in the radial direction, if the basis coefficients decay fast enough to overcome the growth of the basis functions near the domain boundary. We construct general d -dimensional Zernike bases and prove that they provide examples of infinite-dimensional Lipschitz stability.

We provide an elementary proof of the fact that a sequence defined by a linear recurrence relation with integer coefficients is periodic if and only if all characteristic roots are distinct roots of unity. Additionally, we discuss the case in which the coefficients of the recurrence relation are restricted to the set {–1,0,1}.

Kleene Algebra (KA) is a useful tool for proving that two programs are equivalent. Because KA's equational theory is decidable, it integrates well with interactive theorem provers. This raises the question: which equations can we (not) prove using the laws of KA? Moreover, which models of KA are complete, in the sense that they satisfy exactly the provable equations? Kozen (1994) answered these questions by characterizing KA in terms of its language model. Concretely, equivalences provable in KA are exactly those that hold for regular expressions. Pratt (1980) observed that KA is complete w.r.t. relational models, i.e., that its provable equations are those that hold for any relational interpretation. A less known result due to Palka (2005) says that finite models are complete for KA, i.e., that provable equivalences coincide with equations satisfied by all finite KAs. Phrased contrapositively, the latter is a finite model property (FMP): any unprovable equation is falsified by a finite KA. Both results can be argued using Kozen's theorem, but the implication is mutual: given that KA is complete w.r.t. finite (resp. relational) models, Palka's (resp. Pratt's) arguments show that it is complete w.r.t. the language model. We embark on a study of the different complete models of KA, and the connections between them. This yields a novel result subsuming those of Palka and Pratt, namely that KA is complete w.r.t. finite relational models. Next, we put an algebraic spin on Palka's techniques, which yield a new elementary proof of the finite model property, and by extension, of Kozen's and Pratt's theorems. In contrast with earlier approaches, this proof relies not on minimality or bisimilarity of automata, but rather on representing the regular expressions involved in terms of transformation automata.

The author of this article uses demand-and-supply diagrams to study a production economy with a numéraire, providing elementary proofs to fundamental economic results. He demonstrates that the competitive outcome lies within the core of the economy’s market game and illustrates core convergence to the competitive equilibrium when the market thickens. He further explains the impossibility of efficient trade under private preferences and incentive constraints. His graphical approach makes economic theory more accessible and unified for learning and teaching.

A number is normal in base b if, in its base b expansion, all blocks of digits of equal length have the same asymptotic frequency. The rate at which a number approaches normality is quantified by the classical notion of discrepancy, which indicates how far the scaling of the number by powers of b is from being equidistributed modulo 1. This rate is known as the discrepancy of a normal number. The Champernowne constant C 10 = 0.12345678910111213141516 … is the most well-known example of a normal number. In 1986, Schiffer provided the discrepancy of numbers in a family that includes the Champernowne constant. His proof relies on exponential sums. Here, we present a discrete and elementary proof specifically for the discrepancy of the Champernowne constant.

Let { P n } \{P_n\} be the Pell sequence. By combining the congruence properties of recurrence sequences with the law of quadratic reciprocity, it is proved that for odd n n , P n P_n is a perfect square if and only if n = ± 1 , ± 7 n=\pm 1, \pm 7 . This provides an elementary proof for Ljunggren’s result, which asserts that the only positive integer solutions of the Diophantine equation x 2 − 2 y 4 = − 1 x^2-2y^4=-1 are ( x , y ) = ( 1 , 1 ) (x, y)=(1, 1) and ( 239 , 13 ) (239, 13) .

The M-convexity of dual Schubert polynomials was first proven by Huh, Matherne, Mészáros, and St. Dizier in 2022. We give a full characterization of the support of dual Schubert polynomials, which yields an elementary alternative proof of the M-convexity result, and furthermore strengthens it by explicitly characterizing the vertices of their Newton polytopes combinatorially. Using this characterization, we give a polynomial-time algorithm to determine if a coefficient of a dual Schubert polynomial is zero, analogous to a result of Adve, Robichaux, and Yong for Schubert polynomials.

We construct a basis of the Garsia-Procesi ring using the catabolizability type of standard Young tableaux and the charge statistic. This basis turns out to be equal to the descent basis defined in [3]. Our new construction connects the combinatorics of the basis with the well-known combinatorial formula for the modified Hall-Littlewood polynomials H ˜ μ [ X ; q ] , due to Lascoux, which expresses the polynomials as a sum over standard tableaux that satisfy a catabolizability condition. In addition, we prove that identifying a basis for the antisymmetric part of R μ with respect to a Young subgroup S γ is equivalent to finding pairs of standard tableaux that satisfy conditions regarding catabolizability and descents. This gives an elementary proof of the fact that the graded Frobenius character of R μ is given by the catabolizability formula for H ˜ μ [ X ; q ] .

Abstract Inspired by the theory of hyperbolic polynomials and Hodge theory, we develop the theory of Lorentzian polynomials on cones. This notion captures the Hodge-Riemann relations of degree zero and one. Motivated by fundamental properties of volume polynomials of Chow rings of simplicial fans, we define a class of multivariate polynomials which we call hereditary polynomials. We give a complete and easily checkable characterization of hereditary Lorentzian polynomials. This characterization is used to give elementary and simple proofs of the Heron-Rota-Welsh conjecture for the characteristic polynomial of a matroid, and the Alexandrov-Fenchel inequalities for convex bodies. We then characterize Chow rings of simplicial fans which satisfy the Hodge-Riemann relations of degree zero and one, and we prove that this property only depends on the support of the fan. Several different characterizations of Lorentzian polynomials on cones are provided.

In this paper, we consider the Cauchy problem of 3D quasilinear wave equations which satisfy the weak null condition. First, we use Fourier transform to express the solution in terms of the profile function, and then introduce a dyadic decomposition in phase space, and, hence, by applying the space-time resonance method, we give an elementary proof of the modified scattering result of Deng and Pusateri [3]. We also improve the estimate of the error term between the solution of wave equation and the solution of the related asymptotic equation.

The note contains a short elementary proof of Cayley’s formula for labeled trees. We build trees inductively and use an elementary algebra trick to sum over all possible tree types.

Abstract We review notions of mass of asymptotically locally Anti-de Sitter three-dimensional spacetimes, and apply them to some known solutions. For two-dimensional general relativistic initial data sets the mass is not invariant under asymptotic symmetries, but a unique mass parameter can be obtained either by minimisation, or by a monodromy construction, or both. We give an elementary proof of positivity, and of a Penrose-type inequality, in a natural gauge. We carry-out a gluing construction at infinity to time-symmetric asymptotically locally hyperbolic vacuum initial data sets and derive mass/entropy formulae for the resulting manifolds.
Finally, we show that all mass aspect functions can be realised by constant scalar curvature metrics on complete manifolds which are smooth except for at most one conical singularity.

Elementary proofs of recent congruences for overpartitions wherein non-overlined parts are not divisible by 6

We provide a proof of the layer cake representation theorem that requires only basic measure theory (the monotone convergence theorem) without using Fubini’s theorem, thus removing the condition of σ -finiteness.

Abstract We prove that an étale fibration between ‐bundles admits local sections composed of several elementary morphisms of particularly simple and accessible type. As applications, we establish an inverse function theorem for ‐bundles and provide an elementary proof that every weak equivalence of ‐bundles induces a quasi‐isomorphism of the differential graded algebras of global functions. Furthermore, we apply this inverse function theorem to show that the homotopy category of ‐bundles admits a simple description in terms of homotopy classes of morphisms, when ‐bundles are restricted to their germs around their classical loci.

First we provide short, elementary and self-contained proofs of all known results concerning the lower bounds of the Banach-Mazur distances between the space $c_0$ of sequences converging to $0$ and other $\ell_1$-preduals isomorphic to $c_0$. Then, we use our technique to obtain lower bounds for the Banach-Mazur distances between any two $\ell_1$-preduals $X$ and $Y$. Our estimate depends only on the smallest radiuses $r^*(X)$ and $r^*(Y)$ of the closed balls in $\ell_1$ containing, respectively, all $\sigma(\ell_1,X)$-cluster points and all $\sigma(\ell_1,Y)$-cluster points of the set of all extreme points of the closed unit ball in $\ell_1$, and for any values of $r^*(X)$ and $r^*(Y)$ it is sharp. We apply this result to show that for every $\ell_1$-predual $X$ with $r^*(X)< 1$, every $\ell_1$-predual $Y$ with the distance from $X$ strictly less than $\frac{3-r^*(X)}{1+r^*(X)}$ induces a weak$^*$ topology on $\ell_1$ such that $\ell_1$ has the $\sigma(\ell_1,Y)$-fixed point property for nonexpansive mappings. If we additionally assume that the standard basis in $\ell_1$ is $\sigma(\ell_1,X)$-convergent, then the estimate is precise. The same holds if the standard basis in $\ell_1$ has a finite number of $\sigma(\ell_1,X)$-cluster points and each of them has a finite number of non-zero coordinates. It should be emphasized that the value of this constant was known only for the space $c_0$ so far.

Elementary proofs of some congruences for (k, a)-colored F-partitions