Classical number theory has developed many descriptors for integers, including predicates, arithmetic functions, and structural invariants. The majority of them capture a specific aspect of an integer rather than providing a unified representation across multiple mathematical domains. We introduce the Universal Entropy Fingerprint Framework (UEFT), a computational framework for representing integers through multiple Shannon entropy coordinates derived from selected mathematical transformations across five complementary domains: algebraic (prime-exponent structure), geometric (divisor-lattice rank distribution), arithmetic (σ-weighted divisor magnitude), positional (multi-base digit distribution), and dynamical (Collatz trajectory parity). Every coordinate is computed using a single Shannon entropy definition, avoiding the ad hoc mixing of incompatible information measures that has limited previous integer-invariant proposals. We compute the full fingerprint for all integers up to N = 1,000,000 using sieve-based factorization and closed-form combinatorial identities, eliminating the need to explicitly construct divisor graphs or partition objects. The resulting 19-coordinate fingerprint space requires 17 of 19 standardized coordinates to reach 95% cumulative variance under Principal Component Analysis (PCA); this is partly driven by the 15 near-collinear positional coordinates. Exemplar integers spanning primes, prime powers, highly composite, and perfect numbers exhibit distinct fingerprint profiles, illustrating how classical arithmetic categories are represented within a continuous information-geometric space.
We propose truncation-based median ranked set sampling (TBMRSS). The estimator of the population mean and median is studied under this design. We show that the population mean estimator is unbiased for symmetric distributions. We also derive closed-form variances and efficiencies relative to simple random sampling for the uniform, exponential, and normal distributions. For the median estimator, we further show that its asymptotic relative efficiency is free of the parent distribution and is largest for the median-only design. Monte-Carlo studies are carried out over eight symmetric and skewed distributions and for different set sizes, comparing TBMRSS with several variations of ranked set sampling (RSS). The studies show that for the population mean moderate truncation makes TBMRSS the most efficient of the schemes, while for the population median TBMRSS dominates classical median RSS by 22–26%. The gains persist under the fraction- of-random-rankings model of imperfect ranking. An application to the Wisconsin Diagnostic Breast Cancer data confirms these findings on real data.
Wilfredo Bazán-Ramírez, Javier Antonio Prudencio-Vidal, Amador Grover Mejía-Osorio, Edgar Gutiérrez-Gómez, Ybar Gustavo Palomino-Malpartida, Aldo Bazán-Ramírez, Miguel Ángel Bello-Bernal
This study proposes an integrated quantitative framework combining the Altman Z″-score with Binary Segmentation change-point detection—using a least squares cost function—to examine corporate financial distress as a dynamic regime-shifting process amenable to structural analysis, addressing the gap in longitudinal distress monitoring for capital-intensive firms in emerging markets. The framework formally minimizes a piecewise least squares cost function over a financial time series of 56 consecutive quarterly observations (2011Q1–2024Q4), identifying endogenous breakpoints that delimit regimes internally homogeneous in mean level of financial deterioration. Because these breakpoints are selected endogenously, standard Chow test critical values are complemented by a supremum-Wald test accounting for the search over candidate break locations, which confirms the presence of at least one genuine structural break (sup-Wald = 38.61, exceeding the 1% asymptotic critical value). The framework is applied to Telefónica del Perú S. A. A.—a publicly listed capital-intensive telecommunications operator that filed a request to enter insolvency restructuring proceedings in February 2025—providing an empirical case with a formally observable ex post validation event. Five endogenously detected breakpoints—located at 2013Q2, 2014Q3, 2018Q2, 2020Q4, and 2022Q1—delimit six structurally differentiated regimes, with mean Z″-score values evolving from 1.58 (phase 1: moderate vulnerability) to a transitory peak of 2.42 (phase 2: transitory stability peak), followed by a progressive decline to 0.60 (phase 3: transition to distress), −0.29 (phase 4: structural distress onset), −1.40 (phase 5: accelerated deterioration), to −2.52 (phase 6: severe terminal distress). Under the primary market-value specification, persistent financial distress signals were observable from 2015Q2—a horizon that emerges naturally from the longitudinal depth of the analysis and exceeds the anticipatory windows typically documented in prior retrospective studies on corporate financial distress. These findings demonstrate that integrating change-point detection algorithms with accounting-based financial indicators substantially enriches the dynamic interpretation of insolvency risk trajectories beyond static classification approaches. A sensitivity analysis comparing market-value and book-value specifications of the model confirms that the identified structural breakpoints are robust to this methodological choice, although the resulting Z″-score levels and distress-zone classifications are specific to the market-value specification adopted as primary in this study, addressing a gap not systematically tested in prior applied literature. The proposed framework offers a replicable mathematical approach for longitudinal monitoring applicable to comparable capital-intensive firms in emerging markets.
Compound process models are widely used to model the sum of claims against an insurance portfolio or total withdrawals from a bank. The classic compound Poisson-Exponential model is often used, and the standard approach estimates unknown parameters from historical data using maximum likelihood estimation. Unfortunately, this method can significantly underestimate risk. We use a Bayesian approach to compute a predictive distribution for cumulative loss and the ruin probability (i.e., bankruptcy) and obtain closed-form, analytical solutions. We examine the performance of these methods and show, under a range of sample sizes, that our method outperforms traditional approaches, and we provide statistical evidence that it achieves zero coverage error, improving risk assessment.
Mohammad Akram, Umar Ishtiaq, Khaleel Ahmad, Ioan-Lucian Popa
Plug-and-Play methods solve imaging inverse problems by inserting a learned denoiser into a proximal splitting algorithm. Their convergence theory presently requires the denoiser, or its residual, to be nonexpansive or firmly nonexpansive, a constraint that curbs the network and, in turn, the reconstruction quality. We prove that this requirement can be weakened to enriched nonexpansivity, a strictly larger operator class in which the map is nonexpansive only after averaging. Using the displacement and transfer–return calculus for nonexpansive-type operators, we show that when the composite Plug-and-Play operator is enriched nonexpansive an inertial Mann or Ishikawa iteration converges to a fixed point of the operator, that is, to a Plug-and-Play solution, and we identify the inertial correction as the exact quantity that reduces the enriched iteration to a classical iteration of the nonexpansive generator. Around this reduction we build a certification theory. The operator is b -enriched exactly when the displacement I− T is infinitesimally cocoercive with modulus 1/(2(b +1)), so that the smallest admissible constant is one less than the supremum of a Rayleigh-type quotient of that displacement. The characterization is attained on non-scalar operators, is computable by automatic differentiation, and remains bounded in the near-identity regime where the field-of-values bound degenerates. A perturbation estimate converts spectral measurements on a finite sample into a bound over a segment or a covered set, which is what the Fejér argument needs and what a program can carry out. We add a training objective that promotes the certified inequality, a safeguarded step-size controller with a closed-form greedy rule and an amortized policy trained by unrolled differentiation, sublinear and linear rates, stability under inexact denoiser evaluation, and a splitting-agnostic extension covering Plug-and-Play ADMM. Two experiments support the theory. On a linear surrogate, for which every constant is available in closed form, the certificates are confirmed and the exact characterization widens the admissible step-size window by about 31% over the field-of-values bound. On Gaussian deblurring, motion deblurring and single-image super-resolution with trained denoisers, the certified constant is measured for each composite operator. On the deblurring task a trained composite with Lipschitz constant 2.08, admitted by no nonexpansive condition, is certified with constant 0.60, where the characterization is more than fifteen times tighter than the field-of-values bound; it restores within four hundredths of a decibel of a nonexpansive composite of identical architecture, so the larger class is reached at no measured cost in quality. The greedy relaxation halves the iteration count of the constant-step scheme. The acceleration is measured among safeguarded schemes; unrelaxed Picard, which carries no guarantee here, remains faster on the linear instance.
