Let L(Kn) be the line graph of Kn and let HS6 be the graph obtained from a 4-cycle by attaching a pendant edge at each of two antipodal vertices. We prove that L(Kn) has an HS6-decomposition if and only if n≥4 and n≢3(mod4). The proof splits L(Kn) into smaller line graphs together with graphs W(m,k) built from k triangle bundles glued along shared degree-two vertices, characterises the single-copy case, and handles W(m,2) via a construction attached to each cycle of a cycle decomposition of Km.
We consider a compound-Poisson insurance reserve process with stochastic premium income, phase-type distributed claims and premiums and a segmented constant dividend outflow. We demonstrate that the non-local equations of ruin and dividend valuation admit a finite dimensional matrix representation which involves careful considerations of the boundary conditions. We find the free premium states and write the transversality equations. We extend the framework to accommodate discounted Gerber–Shiu penalties depending on both the reserve before ruin and the deficit. We explicitly keep the claim-penalty forcing term, including ruin as a side effect of continuous dividend payout. Instead of considering numerically observed spectral separation as a universal theorem we give a polynomial root-count analysis. The effects of fluctuations of premiums at constant mean, including Erlang-2 claims, are revealed by systematic comparisons and an autonomous numerical execution. We expand on the example of MTPL2: we see earned premiums, but not when the premium is paid. The MTPL2 earned-premium rate is distinguished from the income rate specified for a hypothetical 20% loading scenario. The empirical accounting figures and the stochastic cash-flow experiments are presented separately, since the dataset does not identify payment times or individual receipt sizes.
We study the asymptotic expansion of three-dimensional Laplace-type integrals with a non-stationary boundary minimum point. The presence of two tangential directions introduces additional geometric complexity in the asymptotic analysis. We develop a geometric approach based on the successive combination of a tubular transformation, an angular reduction, and a polar parametrization of the tangential variables, reducing the original three-dimensional problem to a family of direction-dependent two-dimensional Laplace-type integrals. The main contribution is the introduction of a directional order of contact describing the local interaction between the boundary and the level surface of the phase function in each tangential direction. Under suitable regularity assumptions and a constant directional contact order, a complete asymptotic expansion is obtained, together with a remainder estimate uniform with respect to the tangential direction. The leading term has order λ−(1+2/p) and an explicit leading coefficient. The framework covers both the classical non-degenerate case p=2 and degenerate cases with higher even contact order. An application to the Gibbs equilibrium of a confined particle is presented, for which the leading coefficient is computed explicitly and the resulting asymptotic formula is numerically verified.
This study develops a benchmark-relative application of geometric Brownian motion (GBM) for evaluating the capital-value performance of real estate under uncertainty. It introduces a valuation-regime taxonomy that distinguishes the growth conditions for benchmark-adjusted mean, median, and modal property values and links them to benchmark underperformance and downside risk over alternative holding horizons. Using a compiled annual median residential property-price series for Accra and Tema (2016–2026), the Ghanaian application illustrates how nominal property appreciation can coexist with deteriorating benchmark-relative performance when appreciation is insufficient relative to the selected investment benchmark. The analysis further demonstrates that expected and median benchmark-adjusted values may exhibit opposing growth patterns, while the probability of benchmark underperformance depends on the interaction among appreciation, benchmark return, volatility, and investment horizon. Rental-yield sensitivity further shows that relatively small positive net yields can alter the classification. The proposed approach provides a probabilistic assessment of benchmark-relative capital-value performance, complementing conventional nominal growth projections.
This paper presents an improved infeasible interior-point algorithm tailored for semi-definite linear complementarity problems (SDLCPs). The proposed approach leverages full Nesterov–Todd search directions while eliminating centering steps, significantly simplifying the algorithmic structure and reducing computational overhead. A rigorous theoretical analysis confirms that the method is well defined, globally convergent, and achieves an iteration complexity that matches the best-known bounds for infeasible interior-point methods in this domain. Complementing the theoretical results, extensive numerical experiments highlight the algorithm’s superior practical performance, consistently achieving faster convergence with fewer iterations compared to existing methods. These findings demonstrate the algorithm’s potential as an efficient and robust solution technique for SDLCPs, with promising applicability to large-scale and complex problem instances.
