arXiv Math @arxiv_math@qoto.org

In this study, we introduce a refined method for ascertaining error estimations in numerical simulations of dynamical systems via an innovative application of composition techniques. Our approach involves a dual application of a basic one-step numerical method of order p in this part, and for the class of Backward Difference Formulas schemes in the second part [Deeb A., Dutykh D. and AL Zohbi M. Error estimation for numerical approximations of ODEs via composition techniques. Part II: BDF methods, Submitted, 2024]. This dual application uses complex coefficients, resulting outputs in the complex plane. The methods innovation lies in the demonstration that the real parts of these outputs correspond to approximations of the solutions with an enhanced order of p + 1, while the imaginary parts serve as error estimations of the same order, a novel proof presented herein using Taylor expansion and perturbation technique. The linear stability of the resulted scheme is enhanced compared to the basic one. The performance of the composition in computing the approximation is also compared. Results show that the proposed technique provide higher accuracy with less computational time. This dual composition technique has been rigorously applied to a variety of dynamical problems, showcasing its efficacy in adapting the time step,particularly in situations where numerical schemes do not have theoretical error estimation. Consequently, the technique holds potential for advancing adaptive time-stepping strategies in numerical simulations, an area where accurate local error estimation is crucial yet often challenging to obtain.

We use the stacky approach to $p$-adic cohomology theories recently developed by Drinfeld and Bhatt--Lurie to generalise known comparison theorems in $p$-adic Hodge theory so as to accommodate coefficients. More precisely, we establish a comparison between the rational crystalline cohomology of the special fibre and the rational $p$-adic étale cohomology of the arithmetic generic fibre of any proper $p$-adic formal scheme $X$ which allows for coefficients in any crystalline local system on the generic fibre of $X$; moreover, we also prove a comparison between the Nygaard filtration and the Hodge filtration for coefficients in an arbitrary gauge in the sense of Bhatt--Lurie. In the process, we develop a stacky approach to diffracted Hodge cohomology as introduced by Bhatt--Lurie, establish a version of the Beilinson fibre square of Antieau--Mathew--Morrow--Nikolaus with coefficients in the proper case and prove a comparison between syntomic cohomology and $p$-adic étale cohomology with coefficients in an arbitrary $F$-gauge. This work is the author's master's thesis at the University of Bonn.

We give an asymptotic formula for the number of $\mathbb{F}_{q}$-rational points over a fixed determinant moduli space of stable vector bundles of rank $r$ and degree $d$ over a smooth, projective curve $X$ of genus $g \geq 2$ defined over $\mathbb{F}_{q}.$ Further, we study the distribution of the error term when $X$ varies over a family of hyperelliptic curves. We then extend the results to the Seshadri desingularisation of the moduli space of semi-stable vector bundles of rank $2$ with trivial determinant, and also to the moduli space of rank $2$ stable Higgs bundles.

We classify smooth projective varieties of Picard rank 2 which has two structures of blow-up of projective space along smooth subvarieties of different dimensions. This gives a characterization of the so called quadro-cubic Cremona transformation.

Let $\mathbb{F}$ be a field, and consider the hypercube $\{ 0, 1 \}^{n}$ in $\mathbb{F}^{n}$. Sziklai and Weiner (Journal of Combinatorial Theory, Series A 2022) showed that if a polynomial $P ( X_{1}, \dots, X_{n} ) \in \mathbb{F}[ X_{1}, \dots, X_{n}]$ vanishes on every point of the hypercube $\{0,1\}^{n}$ except those with at most $r$ many ones then the degree of the polynomial will be at least $n-r$. This is a generalization of Alon and Füredi's fundamental result (European Journal of Combinatorics 1993) about polynomials vanishing on every point of the hypercube except at the origin (point with all zero coordinates). Sziklai and Weiner proved their interesting result using Möbius inversion formula and the Zeilberger method for proving binomial equalities. In this short note, we show that a stronger version of Sziklai and Weiner's result can be derived directly from Alon and Füredi's result.

Let $R\ $be an integral domain and $R^{\#}$ the set of all nonzero nonunits of $R.\ $For every elements $a,b\in R^{\#},$ we define $a\sim b$ if and only if $aR=bR,$ that is, $a$ and $b$ are associated elements. Suppose that $EC(R^{\#})$ is the set of all equivalence classes of $R^{\#}\ $according to $\sim$.$\ $Let $U_{a}=\{[b]\in EC(R^{\#}):b\ $divides $a\}$ for every $a\in R^{\#}.$ Then we prove that the family $\{U_{a}\}_{a\in R^{\#}}$ becomes a basis for a topology on $EC(R^{\#}).\ $This topology is called divisor topology of $R\ $and denoted by $D(R).\ $We investigate the connections between the algebraic properties of $R\ $and the topological properties of$\ D(R)$. In particular, we investigate the seperation axioms on $D(R)$, first and second countability axioms, connectivity and compactness on $D(R)$. We prove that for atomic domains $R,\ $the divisor topology $D(R)\ $is a Baire space. Also, we characterize valution domains $R$ in terms of nested property of $D(R).$ In the last section, we introduce a new topological proof of the infinitude of prime elements in a UFD and integers by using the topology $D(R)$.

Quantitative measurement of ageing across systems and components is crucial for accurately assessing reliability and predicting failure probabilities. This measurement supports effective maintenance scheduling, performance optimisation, and cost management. Examining the ageing characteristics of a system that operates beyond a specified time $t > 0$ yields valuable insights. This paper introduces a novel metric for ageing, termed the Variance Residual Life Ageing Intensity (VRLAI) function, and explores its properties across various probability distributions. Additionally, we characterise the closure properties of the two ageing classes defined by the VRLAI function. We propose a new ordering, called the Variance Residual Life Ageing Intensity (VRLAI) ordering, and discuss its various properties. Furthermore, we examine the closure of the VRLAI order under coherent systems.

New formulae for a summation over a positive part of $SL(2,\mathbb Z)$ are presented. Such formulae can be written for any convex curve. We present several formulae where $π$ is obtained.

The existence of $srg(99,14,1,2)$ has been a question of interest for several decades to the moment. In this paper we consider the structural properties in general for the family of strongly regular graphs with parameters $λ=1$ and $μ=2$. In particular, we establish the lower bound for the number of hexagons and, by doing that, we show the connection between the existence of the aforementioned graph and the number of its hexagons.

We present a survey of results related to the solution of Kolmogorov--Nikolsky problem for Fourier sums on the classes of generalized Poisson integrals $C^{α,r}_{β,p}$, which consists in finding of asymptotic equalities for exact upper boundaries o f uniform norms of deviations of partial Fourier sums on the classes of $2π$--periodic functions $C^{α,r}_{β,p}$, which are defined as convolutions of the functions, which belong to the unit balls pf the spaces $L_{p}$, $1\leq p\leq \infty$, with generalized Poisson kernels $$ P_{α,r,β}(t)=\sum\limits_{k=1}^{\infty}e^{-αk^{r}}\cos \big(kt-\frac{βπ}{2}\big), \ α>0, r>0, \ β\in \mathbb{R}.$$

In this paper, we have explored the impact of certain indices-dependent element-wise transformations on the null space of a matrix. We have found the conditions on this transformation that will preserve the rank and nullity of the original matrix. We have also found some transformations which give localized null vectors for the transformed matrix. Finally, some possible applications of these localized null vectors and eigenvalues are mentioned in different domains.

We consider the inverse eigenvalue problem of constructing a substochastic matrix from the given spectrum parameters with the corresponding eigenvector constraints. This substochastic inverse eigenvalue problem (SstIEP) with the specific eigenvector constraints is formulated into a nonconvex optimization problem (NcOP). The solvability for SstIEP with the specific eigenvector constraints is equivalent to identify the attainability of a zero optimal value for the formulated NcOP. When the optimal objective value is zero, the corresponding optimal solution to the formulated NcOP is just the substochastic matrix desired to be constructed. We develop the alternating minimization algorithm to solve the formulated NcOP, and its convergence is established by developing a novel method to obtain the boundedness of the optimal solution. Some numerical experiments are conducted to demonstrate the efficiency of the proposed method.

We give a full analytic solution to a particular case of the algebraic Riccati equation $XWW^*WX=W^*$ for any matrix $W$ (possibly non-square or non-symmetric) in using the Schur method, terms of the SVD decomposition of $W$. In particular, $(WX)^3=WX$ and $(XW)^3=XW$ for any solution $X$. We show that for $W=AB$, matrix $X=B^+ A^+$ is a solution of this equation if and only if the reverse order law holds, i.e., ${(AB)}^+=B^+ A^+$. For a Hermitian and invertible $W$ the maximal and stabilizing Hermitian solutions is shown to be equal to $W^+$. Equivalence to the equation $XWX=W^+$ is proven.

This paper conducts a geometric analysis of the Joint-Eigenspace Fourier transform of the symmetric space of the non-compact type. Our study shows how the Poisson transform builds up the well-known Helgason Fourier transform for an analysis of the complete duality of the underlying symetric space. Among other results, we establish an inversion formula, a Plancherel formula and (as our main result) a Paley-Wiener theorem for the Joint-Eigenspace Fourier transform on any noncompact symmetric space.

This article characterizes the associativity of two-place functions $T: [0,1]^2\rightarrow [0,1]$ defined by $T(x,y)=f^{(-1)}(F(f(x),f(y)))$ where $F:[0,1]^2\rightarrow[0,1]$ is a triangular norm (even a triangular subnorm), $f: [0,1]\rightarrow [0,1]$ is a strictly increasing function and $f^{(-1)}:[0,1]\rightarrow[0,1]$ is the pseudo-inverse of $f$. We prove that the associativity of functions $T$ only depends on the range of $f$, which is used to give a sufficient and necessary condition for the function $T$ being associative when the triangular norm $F$ is an ordinal sum of triangular norms and an ordinal sum of triangular subnorms in the sense of A. H. Clifford, respectively. These results finally are applied for describing classes of triangular norms generated by strictly increasing functions.

In this paper, we generalize the notion of joint eigenvalues and joint spectrum of matrices and operator tupples on a bi complex Hilbert space. We observe that unlike the spectrum of a bounded operator on a bi complex Hilbert space is bounded. But this is not the case here. Even the set of joint eigenvalues of matrix tuples is unbounded.

Cooperative perception, offering a wider field of view than standalone perception, is becoming increasingly crucial in autonomous driving. This perception is enabled through vehicle-to-vehicle (V2V) communication, allowing connected automated vehicles (CAVs) to exchange sensor data, such as light detection and ranging (LiDAR) point clouds, thereby enhancing the collective understanding of the environment. In this paper, we leverage an importance map to distill critical semantic information, introducing a cooperative perception semantic communication framework that employs intermediate fusion. To counter the challenges posed by time-varying multipath fading, our approach incorporates the use of orthogonal frequency-division multiplexing (OFDM) along with channel estimation and equalization strategies. Furthermore, recognizing the necessity for reliable transmission, especially in the low SNR scenarios, we introduce a novel semantic error detection method that is integrated with our semantic communication framework in the spirit of hybrid automatic repeated request (HARQ). Simulation results show that our model surpasses the traditional separate source-channel coding methods in perception performance, both with and without HARQ. Additionally, in terms of throughput, our proposed HARQ schemes demonstrate superior efficiency to the conventional coding approaches.

We conducted a reproducibility study on Integrated Gradients (IG) based methods and the Important Direction Gradient Integration (IDGI) framework. IDGI eliminates the explanation noise in each step of the computation of IG-based methods that use the Riemann Integration for integrated gradient computation. We perform a rigorous theoretical analysis of IDGI and raise a few critical questions that we later address through our study. We also experimentally verify the authors' claims concerning the performance of IDGI over IG-based methods. Additionally, we varied the number of steps used in the Riemann approximation, an essential parameter in all IG methods, and analyzed the corresponding change in results. We also studied the numerical instability of the attribution methods to check the consistency of the saliency maps produced. We developed the complete code to implement IDGI over the baseline IG methods and evaluated them using three metrics since the available code was insufficient for this study.

Any choice of a spherical fusion category defines an invariant of oriented closed 3-manifolds, which is computed by choosing a triangulation of the manifold and considering a state sum model that assigns a 6j symbol to every tetrahedron in this triangulation. This approach has been generalized to oriented closed 3-manifolds with defect data by Meusburger. In a recent paper, she constructed a family of invariants for such manifolds parametrised by the choice of certain spherical fusion categories, bimodule categories, finite bimodule functors and module natural transformations. Meusburger defined generalised 6j symbols for these objects, and introduces a state sum model that assigns a generalised 6j symbol to every tetrahedron in the triangulation of a manifold with defect data, where the type of 6j symbol used depends on what defect data occur within the tetrahedron. The present work provides non-trivial examples of suitable bimodule categories, bimodule functors and module natural transformation, all over categories of $G$-graded vector spaces. Our main result is the description of module functors in terms of matrices, which allows us to classify these functors when $G$ is a finite cyclic group. Furthermore, we calculate the generalised 6j symbols for categories of $G$-graded vector spaces, (bi-)module categories over such categories and (bi-)module functors.

Breakthrough Sparsity Theorem, derived independently by Donoho and Elad \textit{[Proc. Natl. Acad. Sci. USA, 2003]}, Gribonval and Nielsen \textit{[IEEE Trans. Inform. Theory, 2003]} and Fuchs \textit{[IEEE Trans. Inform. Theory, 2004]} says that unique sparse solution to NP-Hard $\ell_0$-minimization problem can be obtained using unique solution of P-Type $\ell_1$-minimization problem. In this paper, we derive noncommutative version of their result using frames for Hilbert C*-modules.