arXiv Math @arxiv_math@qoto.org

This paper develops a hierarchical clustering algorithm for weighted projective spaces $\mathbb{P}_{\mathbf{q}}$, utilizing a Finsler metric $ d_F([z], [w]) $ and its rational analogue $ d_{F,\mathbb{Q}}([z], [w])$ to define distances that preserve the non-Euclidean geometry of these quotient manifolds. Defined via geodesic integrals of a scaling invariant Finsler norm weighted by the grades $\mathbf{q} = (q_0, q_1, \dots, q_n)$, these metrics satisfy true metric properties including the triangle inequality, overcoming the limitations of the non-metric dissimilarity measure from prior work.

In this note we first review the concept of D-function, closely connected with Cauchy-Schwarz inequality, and then introduce the notion of P-covariance on a Hilbert space, where $P$ is an orthogonal projection. We show that when P is specialized to be one-dimensional many well-known inequalities such as Buzano, Richard and Walker inequalities are simple consequences of P-covariance inequalities and their relation with D-functions. By means of these concepts enhancements of the previous mentioned inequalities are also established with minimum effort. A more thorough analysis of Walker inequality is presented jointly with some novel financial considerations as well as a new refinement of Holder inequality.

The uncertainty principle is one of the fundamental tools for time-frequency analysis in harmonic analysis, revealing the intrinsic trade-off between time and frequency resolutions. With the continuous development of various advanced time-frequency analysis methods based on the Fourier transform, investigating uncertainty principles associated with these methods has become one of the most interesting topics. This paper studies the uncertainty principles related to the offset linear canonical transform, including the Plancherel-Parseval-Rayleigh identity, the $2p$ order Heisenberg-Pauli-Weyl uncertainty principle and the sharpened Heisenberg-Pauli-Weyl uncertainty principle. Numerical simulations are also proposed to validate the derived results.

In this article, we introduce a new class of coupled fractional Lane-Emden boundary value problems. We employ a novel approach, the fractional Haar wavelet collocation method with the Newton-Raphson method. We analyze the conditions in two cases to present numerical experiments related to the defined system of fractional differential equations. To validate the accuracy of the proposed method we present the convergence of the method, and we demonstrate the method's effectiveness through five numerical experiments, highlighting real-world applications of fractional differential equations. Using figures and tables, we show that the residual error decreases as we increase the value of the maximum level of resolution $J$ while keeping the order of derivatives fixed, and similar trends also observe when $J$ is fixed and vary the order of fractional derivatives. We demonstrate that Mathematica software can be used effectively to solve such nonlinear singular fractional boundary value problems.

In this article, we employ a fractional version of the radius of curvature in Euler's equation for column buckling, enabling us to derive a fractional differential equation in the Caputo sense. We solve this equation and demonstrate that for certain values of the fractional parameter, there exists a critical buckling force. Additionally, we provide a numerical scheme for accurately approximating this critical force.

In this paper, we focus on clarifying the concept of solving equations of degree greater than six using continuous functions or hypergeometric functions and providing another proof of the non-existence of algebraic solutions for equations of degree greater than four. According to the Kolmogorov-Arnold theorem, we will prove that equations of degree greater than five cannot be solved without special conditions between their coefficients using hypergeometric functions. However, we prove that trinomial equations of general form can in general be solved using hypergeometric functions.

This paper establishes new common fixed point theorems for weakly compatible mappings in metric spaces, relaxing traditional requirements such as continuity, compatibility, and reciprocal continuity. We present a unified framework for three self-mappings $T$, $f$, and $g$ with a contractive condition involving a control function $ψ$, along with corollaries extending results to pairs of mappings and upper semi-continuous control functions. Further generalizations include iterated mappings and sequences of mappings. Rigorous examples demonstrate the necessity of hypotheses and show our results strictly generalize theorems by Al-Thagafi \emph{et. al.} \cite{Al-Thagafi2006}, Babu \emph{et. al.} \cite{Babu2007}, Jungck \cite{Jungck1976,Jungck1986}, Singh \cite{Singh1986,Singh1997a}, Som \cite{Som2003}, Song \cite{Song2007} and Zhang \emph{et. al.} \cite{Zhang2008}. Key advancements include eliminating completeness assumptions on the entire space and relaxing mapping compatibility conditions.

We introduce a new class of polylogarithm sums closely related to a family studied by L. Vepštas in 2010. These generalized sums depend on two free parameters and yield closed-form expressions involving the Dirichlet eta function. Additionally, we present an alternative proof for a hyperbolic sum originally discussed by Ramanujan.

We investigate the conditions under which astheno-Kähler structures can be identified on the product of two compact trans-Sasakian manifolds of dimensions greater than 2.

We introduce radial variants of the Wijsman and Attouch-Wets topologies for the family $\mathcal{S}_{rc}^d$ of star sets $A \subseteq \mathbb{R}^d$ that are radially closed.These topologies give rise to new types of convergence for star-shaped sets with respect to the origin, even when such sets are not closed or bounded. Our approach relies on a new family of functionals, called \textit{radial distance functionals}, which measure ``radial distances'' between points $x \in \mathbb{R}^d$ and sets $A \in \mathcal{S}_{rc}^d$. These are natural radial analogues of the classical distance functionals. We prove that our radial Wijsman type topology $τ_{W^r}$ is not metrizable on $\mathcal{S}_{rc}^d$, while our radial Attouch-Wets type topology $τ_{AW^r}$ is completely metrizable. A corresponding radial Attouch-Wets distance $d_{AW^r}$ is introduced, and we prove that $d_{AW}(A,K) \leq d_{AW^r}(A,K)$ for all closed $A,K \in \mathcal{S}_{rc}^d$, where $d_{AW}$ denotes the Attouch-Wets distance. Among others, these results are applied to prove the continuity of the star duality on $\mathcal{S}_{rc}^d$ with respect to both $τ_{W^r}$ and $τ_{AW^r}$, and to establish topological properties of the family of flowers associated with closed convex sets containing the origin.

Let $p$ be an odd prime. Let $K/\mathbb{Q}_p$ be a finite unramified extension. Let $ρ: G_K \to GL_2(\overline{\mathbb{F}}_p)$ be a continuous representation. We prove that $ρ$ has a crystalline lift of small irregular weight if and only if it has multiple crystalline lifts of certain specified regular weights. The inspiration for this result comes from work of Diamond-Sasaki on geometric Serre weight conjectures. Our result provides a way to translate results currently formulated only for regular weights to also include irregular weights. The proof uses results on Kisin and $(φ,\hat{G})$-modules obtained from extending recent work of Gee-Liu-Savitt to study crystalline liftability of irregular weights.

Let $b$ be an integer greater than or equal to $2$. For any integer $n\in \left[b^{λ-1}, b^λ-1\right]$, we denote by $R_λ(n)$ the reverse of $n$ in base $b$, obtained by reversing the order of the digits of $n$. We establish a Bombieri-Vinogradov type theorem for the set of the reverses of the prime numbers. Combined with sieve methods, this permits us to prove that there exist $Ω_b\in\mathbb{N}$ and $c_b>0$ such that, for at least $c_b b^λ λ^{-2}$ primes $p\in \left[b^{λ-1}, b^λ-1\right]$, the reverse $R_λ(p)$ has at most $Ω_b$ prime factors. Some explicit admissible values of $Ω_b$ are given.

In this paper, the notion of pronormal L-subgroups of an L-group has been introduced by using the concept of conjugate L-subgroup. The notion of pronormal L-subgroups has been investigated in context of normality and subnormality of L-subgroups and several related properties have been established. Moreover, the relation of pronormality with normalizers and maximal L-subgroups has been explored.

This paper builds a cumulative tower of Grothendieck universes that provides a precise size discipline for higher type theory. Starting from an increasing sequence of inaccessible cardinals, we give an inductive-recursive definition of universe codes, their decoding functor, and a rank that controls the growth of each code. For every level of the tower we prove closure under dependent products, dependent sums, identity types, and all finite limits and colimits; the colimit part is obtained with a rank-stable quotient constructor. A universe-lifting operation shows strict cumulativity across levels. Assuming propositional resizing at one level, we construct an explicit left adjoint to the inclusion of minus-one truncated types and show that resizing automatically lifts to every higher level. These ingredients combine in an Existence Theorem stating that, over Zermelo-Fraenkel set theory with the chosen inaccessible cardinals, the tower delivers a sound set-theoretic model of higher type theory. The results supply the size infrastructure required for forthcoming work on Rezk completions and higher topos models.

Embedded point spectra of rank one singular perturbations of an arbitrary self-adjoint operator A on a Hilbert space H is studied. It is shown that these perturbations can be regarded as self-adjoint extensions of a densely defined closed symmetric operator B with deficiency indices (1; 1). Assuming the deficiency vector of B is cyclic for its self-adjoint extensions, we prove that the spectrum of A contains a dense Gδ subset where it is not possible to have eigenvalues for any rank one singular perturbation. Moreover, for a dense Gδ set of rank one singular perturbations of A their eigenvalues are isolated. The approach presented here unifies points of view taken by different authors.

This paper addresses the challenging and interesting inverse problem of reconstructing the spatially varying dielectric constant of a medium from phaseless backscattering measurements generated by single-point illumination. The underlying mathematical model is governed by the three-dimensional Helmholtz equation, and the available data consist solely of the magnitude of the scattered wave field. To address the nonlinearity and severe ill-posedness of this phaseless inverse scattering problem, we introduce a robust, globally convergent numerical framework combining several key regularization strategies. Our method first employs a phase retrieval step based on the Wentzel--Kramers--Brillouin (WKB) ansatz, where the lost phase information is reconstructed by solving a nonlinear optimization problem. Subsequently, we implement a Fourier-based dimension reduction technique, transforming the original problem into a more stable system of elliptic equations with Cauchy boundary conditions. To solve this resulting system reliably, we apply the Carleman convexification approach, constructing a strictly convex weighted cost functional whose global minimizer provides an accurate approximation of the true solution. Numerical simulations using synthetic data with high noise levels demonstrate the effectiveness and robustness of the proposed method, confirming its capability to accurately recover both the geometric location and contrast of hidden scatterers.

Classical Finite Volume methods for multi-dimensional problems include stabilization (e.g. via a Riemann solver), that is derived by considering several one-dimensional problems in different directions. Such methods therefore ignore a possibly existing balance of contributions coming from different directions, such as the one characterizing multi-dimensional stationary states. Instead being preserved, they are usually diffused away by such methods. Stationarity preserving methods use a better suited stabilization term that vanishes at the stationary state, allowing the method to preserve it. This work presents a general approach to stationarity preserving Finite Volume methods for nonlinear conservation/balance laws. It is based on a multi-dimensional extension of the global flux approach. The new methods are shown to significantly outperform existing ones even if the latter are of higher order of accuracy and even on non-stationary solutions.

The graphical operation of insplitting is key to understanding conjugacy of shifts of finite type (SFTs) in both one and two dimensions. In this paper, we consider two approaches to studying 2-dimensional SFTs: textile systems and rank-2 graphs. Nasu's textile systems describe all two-sided 2D SFTs up to conjugacy, whereas the 2-graphs (higher-rank graphs of rank 2) introduced by Kumjian and Pask yield associated C*-algebras. Both models have a naturally-associated notion of insplitting. We show that these notions do not coincide, raising the question of whether insplitting a 2-graph induces a conjugacy of the associated one-sided 2-dimensional SFTs. Our first main result shows how to reconstruct 2-graph insplitting using textile-system insplits and inversions, and consequently proves that 2-graph insplitting induces a conjugacy of dynamical systems. We also present several other facets of the relationship between 2-graph insplitting and textile-system insplitting. Incorporating an insplit of the bottom graph of the textile system turns out to be key to this relationship. By articulating the connection between operator-algebraic and dynamical notions of insplitting in two dimensions, this article lays the groundwork for a C*-algebraic framework for classifying one-sided conjugacy in higher-dimensional SFTs.

For a quadratic form $φ$ over a field of characteristic different from $2$, we study whether its group of proper projective similitudes ${\bf PSim}^+(φ)$ is rationally connected (i.e. $R$-trivial). We obtain new sufficient conditions in terms of structure properties of $φ$. We further provide new examples of quadratic forms $φ$ belonging to a given power of the fundamental ideal in the Witt ring and such that ${\bf PSim}^+(φ)$ is not rationally connected.

Given a compact semisimple Lie group G and a maximal torus T of G, we give an explicit description of all left and Ad(T)-invariant pluriclosed Hermitian structures on G in terms of the corresponding root system. They depend on 2d+1 parameters in the irreducible case, where dim(T)=2d.