Department of Mathematics

Among the class of generalized Fourier transformations, the linear canonical transform is of pivotal importance mainly due to its higher degrees of freedom in lieu of the conventional Fourier and fractional Fourier transforms. This article is a continuation of the recent works on the linear canonical Dunkl transforms carried out in Ghazouani et al. (J Math Anal Appl 449:1797–1849, 2017), Mejjaoli (J Pseudo-Differ Oper Appl 16:1–43, 2025). Building upon this, we will introduce and study in this paper the generalized wavelet transform associated with the LCDT, called the Dunkl linear canonical wavelet transform. Then we will formulate several weighted uncertainty principles for this new transformation.

We examine the Sobolev space associated with the linear canonical Dunkl transform and explore some properties of the linear canonical Dunkl operators. Building on these results, we establish a real Paley–Wiener theorem for the linear canonical Dunkl transform. Further, we characterize the square-integrable function f whose linear canonical Dunkl transform of the function is supported in the polynomial domain. Finally, we develop the Boas-type Paley–Wiener theorem for the linear canonical Dunkl transform.

Tikhonov regularization gained a wide popularity in the context of Statistical inverse problems. In this article, we investigate and analyze fractional Tikhonov regularization scheme, which is in fact one of many generalizations of Tikhonov regularization, in learning theory. We delve into the theoretical framework of the fractional scheme to prove the consistency of the algorithm, and the convergence analysis of the solution. Furthermore, optimal rate of convergence has been established under Hölder source condition. A discussion on the rate of convergence is pursued with well established schemes in literature. Numerical experiment carried out through the academic data corroborate our analysis.

Let G be a simple graph and It(G) denote the t-path ideal of G. It is well known that the Castelnuovo–Mumford regularity reg(R/It(G)) and the projective dimension pd(R/It(G)) are bounded below by the quantities (t-1)νt(G) and the big height bight(It(G)), respectively, where νt(G) denotes induced matching number of the hypergraph corresponding to It(G). We show that if t≥4, then the difference between reg(R/It(G)) and (t-1)νt(G), and the difference between pd(R/It(G)) and bight(It(G)) can be arbitrarily large even if we take G to be a tree. This, in particular, disproves a conjecture in Hang and Vu (Graphs Combin 41(1):18, 2025). However, when t=3 and G is chordal, we show that reg(R/I3(G))=2ν3(G) and pd(R/I3(G))=bight(I3(G)), extending the well-known formulas for the edge ideals of chordal graphs. As a consequence, we get that the 3-path ideal of a chordal graph is Cohen–Macaulay if and only if it is unmixed. Additionally, we show that the Alexander dual of I3(G) is vertex splittable when G is a tree, thereby resolving the t=3 case of a recent conjecture in Abdelmalek et al. (Int J Algebra Comput 33(3):481–498, 2023). Also, for each t≥3, we give examples of chordal graphs G such that the duals of the corresponding t-path ideals are not vertex splittable. Furthermore, we extend the formula of the regularity of 3-path ideals of chordal graphs to all t-path ideals of caterpillar graphs.

In this paper, we prove the upper bound conjecture proposed by Saeedi Madani and Kiani on the Castelnuovo–Mumford regularity of generalized binomial edge ideals. We give a combinatorial upper bound of regularity for generalized binomial edge ideals, which is better than the bound claimed in that conjecture. Also, we show that the bound is tight by providing an infinite class of graphs.

In many computational problems, using the Markov Chain Monte Carlo (MCMC) can be prohibitively time-consuming. We propose MCMC-Net, a simple yet efficient way to accelerate MCMC via neural networks. The key idea of our approach is to substitute the true likelihood function of the MCMC method with a neural operator based surrogate. We extensively evaluate the accuracy and speedup of our method on three different partial differential equation-based inverse problems where likelihood computations are computationally expensive, namely electrical impedance tomography, diffuse optical tomography, and quantitative photoacoustic tomography. MCMC-Net performs similar to the classical likelihood counterpart but with a significant speedup. We conjecture that the method can be applied to any problem with a sufficiently expensive likelihood function. We also analyze MCMC-Net in a theoretical setting for the different use cases. We prove a universal approximation theorem-type result to show that the proposed network can approximate the mapping resulting from forward model evaluations to a desired accuracy. Furthermore, we establish convergence of the surrogate posterior to the true posterior under Hellinger distance.

This paper considers the inference of unknown parameters for an extension of the new Pareto-type distribution based on progressive type-II censored data. First, the estimation of the model parameter using maximum likelihood and Bayesian methods has been discussed. The approximated confidence intervals and Bayesian credible intervals are discussed as well. We then establish a Bayesian optimal design with respect to variance minimization criteria. Monte Carlo simulations are implemented to compare different methods of estimation, and finally, two real data sets, where the first one represents the remission times (in months) of bladder cancer patients and the second one represents the repair times (in hours) for an airborne communication transceiver have been analyzed for illustrative purposes.

This paper considers estimating unknown parameters of a new Pareto-type distribution based on Type-I hybrid censoring. First, maximum likelihood and Bayesian method, under squared error and LINEX loss functions, are applied for estimating the parameters involved. The highest posterior density intervals are discussed as well. We establish optimum censoring schemes with respect to cost function optimality criteria. Monte Carlo simulations are implemented to compare different methods and finally, a real data set representing the duration of remission of leukemia patients who were treated by a specific drug is analyzed for illustrative purposes.

In this paper, we consider inference upon unknown parameters of the family of inverted exponentiated distributions when it is known that data are doubly censored. Maximum likelihood and Bayes estimates under different loss functions are derived for estimating the parameters. We use Metropolis-Hastings algorithm to draw Markov chain Monte Carlo samples, which are used to compute the Bayes estimates and construct the Bayesian credible intervals. Further, we present point and interval predictions of the censored data using the Bayesian approach. The performance of proposed methods of estimation and prediction are investigated using simulation studies, and two illustrative examples are discussed in support of the suggested methods. Finally, we propose the optimal plans under double censoring scheme.

In this paper, we consider analysis of a competing risks model using binomial removal based on progressive censored data when failure modes are only partially observed. The latent lifetimes of competing risks follow inverted exponentiated exponential distributions with different shape parameters and a common scale parameter. We explore the study by estimating all unknown parameters using classical and Bayesian techniques. First, we obtain maximum likelihood estimates (MLEs) of model parameters. Subsequently, interval estimates are derived based on the observed Fisher information matrix. We obtain Bayesian estimates using squared error and linear exponential (LINEX) loss functions. The highest posterior density (HPD) intervals are also obtained. We examine impact of removal probability p on the expected experiment time (EET) under progressively censored data. We conduct extensive simulation study to evaluate the performance of all estimators. Numerical illustrations are presented from application viewpoints.

We present two distinct families of imaginary biquadratic fields, each of which contains infinitely many members, with each member having large class groups. Construction of the first family involves elliptic curves and their quadratic twists, whereas to find the other family, we use a combination of elliptic and hyperelliptic curves. Two main results are used, one from Soleng [13] and the other from Banerjee and Hoque [3].

In this note, we connect the n-torsions of the Picard group of an elliptic surface to the n-divisibility of the class group of torsion fields for a given integer n>1. We also connect the n-divisibility of the Selmer group to that of the class group of torsion fields.

In this short note we prove that an involution on certain examples of surfaces of general type with pg=0, acts as identity on the Chow group of zero cycles of the relevant surface. In particular we consider examples of such surfaces when the quotient is an Enriques surface and show that the Bloch conjecture holds for such surfaces.

In this note, we define the notion of Tate–Shafarevich group and Selmer group of the Chow group of zero cycles of degree zero of an abelian variety defined over a global function field and prove that the Selmer group is finite.

Let n be an integer co-prime to 3 and let Zn be the ring of integers modulo n. In this article, we study the structure and generators of the unit group of ZnC3. Further, if Tm denotes the elementary abelian 3-group of order 3m, then we provide the structure of U(ZnTm). We also solve the normal complement problem in each case.

Let G be a finite group and r be a prime divisor of the order of G. An irreducible character of G is said to be quasi r-Steinberg if it is non-zero on every r-regular element of G. A quasi r-Steinberg character of degree is said to be weak r-Steinberg if it vanishes on the r-singular elements of In this article, we classify the quasi r-Steinberg cuspidal characters of the general linear group Then we characterize the quasi r-Steinberg characters of and Finally, we obtain a classification of the weak r-Steinberg characters of

Assume F is a finite field of order pf and q is an odd prime for which pf-1=sqm, where m≥1 and (s,q)=1. In this article, we obtain the order of the symmetric and the unitary subgroup of the semisimple group algebra FCq. Further, for the extension G of Cq=⟨b⟩ by an abelian group A of order pn with CA(b)={e}, we prove that if m>1, or (s+1)≥q and 2n≥f(q-1), then G does not have a normal complement in V(FG).

Fusion frames are widely studied for their applications in recovering signals from large data. These are proved to be very useful in many areas, such as, distributed processing, wireless sensor networks, packet encoding. Inspired by the work of Bemrose et al.[12], this paper delves into the properties and characterizations of weaving fusion frames.

Two frames fn∞n=1 and gn∞n=1 in a separable Hilbert space H are said to be weaving frames, if for every s σ N, fnnσ Ugnneσ is a frame for H. Weaving frames are proved to be very useful in many areas, such as, distributed processing, wireless sensor networks, packet encoding and many more. Inspired by the work of Bemrose et al.[11], this paper delves into the properties and characterizations of weaving frames and weaving Riesz bases.

In this paper we study various properties and characterizations of weaving frames in Krein spaces. In support of our findings, several examples and counter examples are provided, illustrating the applicability of the theoretical results. Additionally, we extend the discussion to an important application in probabilistic erasure, highlighting how weaving frames can be used to mitigate data loss in such scenarios.