Zlatko Pangarić

Let F(n, k) = n^k for nonnegative integers n and k. We study the bivariate finite-difference array generated by the commuting operators Δ_n and Δ_k. The n-difference layers form a finite truncated structure whose support is a stepped array, whose terminal cell is a factorial, and whose boundary values reproduce the Stirling numbers of the second kind scaled by r!. In the k-direction, the same matrix has an eigenfunction structure: Δ^s_k F(n, k) = (n − 1)^sn^k. We define the truncated difference pyramid precisely, derive an explicit falling-factorial expansion for every layer, determine its degree and leading coefficient, and introduce mixed differences Δ^r_n Δ^s_kF that connect the two regimes, for which we give an exact closed form. The work is primarily structural and computational. Its contribution is the definition and systematic analysis of a single bivariate object that unifies two classical one dimensional difference regimes, together with an exact computational protocol that verifies three representations of every cell: the two-parent recursion, the alternating-sum formula, and the Stirling expansion. We report explicit verification for N = 6 and N = 8 with the number of tested cells. The construction extends to any polynomial sequence with a known falling-factorial expansion; further extensions (Sheffer sequences, fractional differences) are outlined but not developed.

This paper introduces the formal framework of Symbolic Structures of Differences (SSD) as a novel approach to the analysis of seismic time series, aiming to provide early warning prior to the occurrence of a main shock. Unlike classical early warning systems based on P-wave detection, the SSD methodology identifies changes in the local geometry of geological deformation through the symbolic encoding of three-point differential structures. Each sample triplet (xk,xk+1,xk+2) is assigned a symbolic structure based on the signs of the first and second differences, generating a space of 27 possible local geometries. From the distribution of these structures, the following metrics are derived: SSD entropy (Esds), symbolic space activity (κ), transition entropy (ε), and the Relational SSD Coefficient (RSC). Preliminary retrospective analysis of data for five significant seismic events — Parkfield 2004 (M6.0), L'Aquila 2009 (M6.3), Tohoku 2011 (M9.0), and the Ridgecrest 2019 sequence (M6.4/M7.1) — shows statistically significant changes in SSD parameters within a time window of 47 to 89 seconds before the arrival of the P-wave. Hybrid systems combining SSD detection with classical P-wave analysis potentially offer superior warning time and accuracy compared to traditional approaches. We caution that the presented numerical results are based on a preliminary analysis of a small sample and require validation on an expanded dataset before any potential operational application.

We introduce Symbolic Structures of Differences (SSD), a method for quantifying the complexity of time series data based on the local geometry of second-order differences. Unlike global entropy measures, SSD captures the diversity of local sequential patterns by analyzing the signs of first and second-order differences within overlapping triplets, mapping them to a space of 27 unique symbols. We provide a theoretical analysis of SSD, proving its invariance under affine transformations and establishing its relationship to permutation entropy. The method's statistical properties, including robustness to noise and finite-size effects, are examined through Monte Carlo simulations. We validate SSD on a benchmark of synthetic and real-world physiological time series, comparing its performance against four established complexity measures (permutation entropy, sample entropy, Lempel-Ziv complexity, and spectral entropy) in the context of detecting epileptic seizures from EEG data. The results demonstrate that SSD offers a competitive and computationally efficient framework for characterizing dynamical regimes and identifying phase transitions, with unique sensitivity to local geometrical structures.

This paper investigates the application of the Symbolic Difference Structure (SSD) method as an innovative tool for multidimensional analysis and classification of elements in the periodic system. By applying the SSD algorithm ($k=3$) to five key physicochemical properties—atomic mass, electronegativity, first ionization energy, number of unpaired electrons, and melting point—the study demonstrates that the periodic system exhibits a recognizable numerical geometry that can be systematically quantified. The results show that SSD classification does not alter established chemical periodicity but enables: (1) quantitative discrimination between genuine nuclear anomalies and metrological artifacts, (2) identification of characteristic “hotspots’’ where multiple properties change simultaneously, and (3) a heuristic framework for predicting properties of heavy and superheavy elements. The analysis is extended using monoisotopic masses and correlation studies, confirming the diagnostic value of the method and opening new pathways for comparative analysis in materials science and chemical education.

This paper presents a new methodological approach to the analysis of numerical sequences that are commonly considered random. This includes the decimal expansion of the number π, stock market indices (e.g., Belex15), pseudorandom numbers (PRNG), cryptographically secure pseudorandom numbers (CSPRNG), physical random number generators (RNG), and quantum random numbers (QRNG). The core method is based on hierarchical computation of higher-order differences and symbolic transformation of signs, enabling structural encoding of each sequence into a symbolic space. The primary objective is to determine whether the decimal expansion of π and related sequences exhibit the same distribution of symbolic patterns as the theoretical model of variations with repetition. The analysis is extended to sequences of 4 and 5 digits, including higher-order differences such as third and fourth order. The results show that empirical distributions of these multilayer structures in the digits of π closely correspond to theoretical distributions derived from all possible variations with repetition. This method opens new possibilities for applications in number theory, cryptography, statistics, and classification of algorithmically generated sequences.