Computer Science and Mathematics

Abstract: This paper presents a self-contained, constructive axiomatic formalization of infinitesimal calculus, termed \textit{Constructive Differential Calculus} (CDC). We highlight the pedagogical and epistemological rift introduced by the rigid $\epsilon-\delta$ formalization of Cauchy and Weierstrass. To restore Leibniz's original geometric intuition, CDC replaces the non-constructive machinery of classical Non-Standard Analysis (such as free ultrafilters) with a minimalist algebraic framework based on basic differential generators. In this revised framework, the polynomial nature of differentials is not postulated but emerges as a theorem of the functional application of the difference operator. Classical results are concisely proved in the algebraic framework of CDC, and their relationship with asymptotic and other constructive approaches is evidenced. Finally, it is shown how this architecture seamlessly extends to multi-variable calculus and higher-order derivatives, rigorously redefining total differentials and partial derivatives as macroscopic projections of intrinsic differential components. The operational superiority of CDC in physical domains is demonstrated by treating derivatives inside shock layers as computable transfinite sequences; this approach naturally recovers classic jump conditions without appealing to test functions or generalized Dirac measures, thereby bridging the gap between strong and weak solutions. The notion of \textit{CDC Renormalization} is defined, which demonstrates that physical divergences and self-energy singularities can be structurally internalized and exactly canceled as matching polynomial scales within the non-archimedean field $\mathcal{D}^+$. By replacing the heuristic subtraction of logical infinities with a native, term-by-term algebraic alignment, the Shadow operator acts as an intrinsic projection of observability, establishing CDC as a rigorous alternative to distributional and non-constructive regularizations in mathematical physics.

Abstract: In this paper, we present a third partial solution for the inverse problem of cardinality calculation of the set of fractals for its subcategory of the virtual fuzzy ones in $n$-dimensional Euclidean space. Consistent with the results of the previous deterministic and random cases, we utilize Mandelbrot’s fundamental definition to address the inverse problem of realizing a level-indexed fuzzy fractal profile. We prove the existence of aleph-two virtual fuzzy fractals with a Hausdorff dimension of a bivariate function of them and the given Lebesgue measure at the core level. The construction successfully employs both weak admissible distance kernels and decorated membership functions to establish these results for fuzzy fractals. Furthermore, we identify an explicit regularity-cardinality dichotomy where the resulting family size collapses to aleph-one under the strict constraint of upper semicontinuity. The problem remains open for other fractal dimensions, iterated fuzzy set systems, and non-Euclidean abstract fuzzy fractal spaces.

Abstract: We introduce a novel, non-Cantorian framework for transfinite mathematics by embedding the hierarchy of ordinal numbers into a structured class \( \mathcal{D}^+ \) of \emph{definitely nonzero} hyperreal sequences. Within this algebraic setting, the countable ordinals are embedded, but hyperreals possess distinct asymptotic growth rates that algebraic operations can fully capture. This approach provides a unification of the different types of mathematical infinities, from Leibniz's differentials and Euler's infinity, on which he discovered the exponential series, to the asymptotic infinities of Landau notation and Hardy fields, up to Cantor's ordinals and cardinals, and Robinson's hyperreals of non-standard analysis. A theorem about the relationship between differentials and continuum is proved, and the implication of this result for non-Cantorian sets is discussed.

Abstract: This paper develops a Fourier-analysis framework induced by proportional arithmetic on the positive real line. Starting from the proportional operations ⊕,⊖,⊙,⊘, we construct proportional functions, proportional differentiation and integration, proportional complex scalars, and a proportional exponential kernel. These ingredients define a proportional Fourier transform by means of proportional multiplication, proportional integration and a proportional oscillatory kernel. The associated representative is then used to prove a correspondence theorem showing compatibility with the classical Fourier transform; this theorem acts as a proof mechanism while the definitions remain internal to the proportional algebra. We define complex-valued proportional Lq, Schwartz and Sobolev spaces, prove Riemann–Lebesgue, inversion, Plancherel and unitary extension results, and derive operational rules for translation, modulation, scaling, differentiation and convolution. The framework is illustrated through proportional Gaussian-type functions, a genuinely complex proportional transform pair, a proportional resolvent equation and a proportional heat equation. The resulting theory is isomorphic to the classical Fourier theory through the logarithmic representative, but it keeps the scalar field, integration, kernel, convolution, and function spaces in proportional form. In this sense, the paper provides a systematic proportional-arithmetic formulation of the Fourier package on the positive real line rather than a new transform outside classical harmonic analysis.

Abstract: Let Ω ⊂ R^N, N ≥ 2, be a bounded connected domain, and let u(t) = e^t∆_D 1 be the Dirichlet heat evolution of the constant initial temperature. We prove that if the exterior normal derivative ∂_νu(·,t_j) is constant on ∂Ω along a sequence t_j ↓ 0, then Ω is a ball, assuming only ∂Ω ∈ C². This answers, in the classical curvature regime, the finite-regularity question posed by Cavallina and Pinamonti for their short-time discrete-flux theorem. The key result is the uniform expansion ∂_νu(y,t) = −1(πt)^-½ + H(y)/2 + o(1) as t ↓ 0, where H is the sum of the principal curvatures with respect to the inward unit normal. More quantitatively, the remainder is bounded by C(√t + ω_H(C√t)) , with ω_H the modulus of continuity of H. If ∂Ω ∈ C^2,α, this gives the rate O(t^α/2). The proof uses a boundary-layer parametrix, ambient Euclidean mollification of a normal extension of H, and an L^∞-to-C¹ estimate for the Dirichlet heat semigroup. The ambient regularization avoids differentiating the metrics of parallel hypersurfaces and therefore does not require third derivatives of the boundary-layer parametrix, ambient Euclidean mollification of a normal extension of H, and an L^∞-to-C¹ estimate for the Dirichlet heat semigroup.

Abstract: Sri Lanka's information technology (IT) workforce is increasingly exposed to digitally intensive work conditions, including prolonged screen time, high cognitive demand, irregular work schedules and blurred work–life boundaries. Although occupational well-being has been discussed in general terms, locally grounded and IT-specific lifestyle health assessment models remain limited. This study developed and empirically examined a digitally driven lifestyle health assessment framework based on the six pillars of lifestyle medicine: nutrition, physical activity, stress management, sleep, social connection and avoidance of harmful substance use. A quantitative cross-sectional survey was administered online to IT professionals in Sri Lanka, and 481 valid responses were analysed. Likert-scale responses were processed using R-based statistical procedures, including reverse scoring, composite index construction, reliability inspection, variance inflation factor testing, correlation analysis, regression modelling, sensitivity analysis and group comparisons. The proposed Healthy Lifestyle Index (HLI) recorded a mean score of 33.76 out of 100, while the transformed WHO-5 well-being score recorded a mean of 31.47 out of 100. HLI showed a moderate, statistically significant positive association with well-being (r = 0.471, p < 0.001), and simple regression indicated that HLI explained 22.2% of the variance in well-being. Workplace arrangement, weekly working hours, education level, marital status and experience were associated with differences in lifestyle health, while job role and company size showed weaker differences. The findings demonstrate the need for digital, data-driven lifestyle analytics to support sustainable workforce well-being in Sri Lanka's IT industry and provide a foundation for future AI-enabled occupational health decision support.

Abstract: We study the extremal behavior of real two-term linear combinations of third-order Zernike modes on the closed unit disk D². These modes arise naturally in Zernike expansions of optical wavefront aberrations. For each of the six unordered pairwise linear combinations of third-order modes, we classify the interior local extrema in terms of the two real coefficients. The trefoil–trefoil case is treated more generally through linear combinations of primary n-multifoils; harmonicity and the maximum principle show that no interior local extrema occur and that absolute extrema are attained on the boundary circle. For the remaining pairwise combinations, we give analytic conditions for the existence, uniqueness, location, and values of local extrema, including degenerate exceptional cases. We also compare these local extrema with boundary values and describe the associated absolute-extremum problem on the boundary circle. Symbolic computations are included in the appendix to document several algebraic reductions, and numerical illustrations are provided to visualize the resulting classifications.

Abstract: Maz'ya's Problem 54 proposes a logarithmic classification of an isolated boundary singularity for the planar equation \[ \Delta u+\alpha(x,y)u_x^2+\beta(x,y)u_y^2=0, \] when the coefficients are smooth and strictly positive in a smooth bounded domain and the Dirichlet trace vanishes away from one boundary point. We show that this conclusion fails under the literal interior assumptions. An elementary composition identity converts a positive harmonic function into a solution of a Riccati-type equation with a positive coefficient. Applied to the Poisson kernel, it produces nontangential power-law blow-up of every order $\gamma\in(0,1)$. In a disk tangent to the singular point the construction is explicit: \[ \begin{aligned} u(x,y)&=\left(\frac{2x}{x^2+y^2}\right)^\gamma-1,\\ \alpha(x,y)=\beta(x,y)&=\frac{1-\gamma}{\gamma} \left(\frac{2x}{x^2+y^2}\right)^{-\gamma}. \end{aligned} \] Here the coefficients belong to $C^\infty(\Omega)$, are strictly positive and globally bounded, but have no positive lower bound and no limit at the singular boundary point. The solution grows like a positive power and therefore cannot have a logarithmic profile. The result concerns only the printed interior formulation; it does not settle any strengthened problem imposing boundary extension and quantitative nondegeneracy of the coefficients.

Abstract:

Based on the isomorphic algebraic structures of the 2D Euclidean field of complex vectors V_ℂ and the field of complex numbers ℂ, in terms of identical geometric products of the elements of both fields, this paper, in the first three sections, brings the algebraic structure of a 3D field of complex vectors, as well as the corresponding fundamental integral identities in those vector fields. Additionally, in the fourth section, fundamental vector relations are presented, which relate to a compact Hermitian manifold embedded in an ambient 3D field of complex vectors.

Abstract: We study a multidimensional Cauchy-type integral equation of Maz’ya type whose singular denominator is governed by scalar functions ϕ(x) and ψ(y). Using the coarea formula and a coarea-principal-value interpretation of the singular integral, the associated operator is factored as K=AϕBψ, where Aϕ is a lifting operator and Bψ is a finite Cauchy transform of a level-set coarea trace. For λ≠0, the original equation is equivalent to a reduced one-dimensional equation λh−Sh=Bψf, followed by the reconstruction u=(f+Aϕh)/λ. Compact Fredholm solvability is obtained in two geometrically regular regimes: separated essential ranges of ϕ and ψ, and overlapping ranges satisfying rank-two transversality together with Hölder regularity of the two-variable coarea density. In the diagonal case ψ=ϕ, the reduction gives a weighted finite Hilbert transform equation, so the correct Fredholm framework is classical one-dimensional singular-integral theory rather than compact-operator theory. The results give a rigorous partial framework under restrictive geometric hypotheses, clarify the exact mechanism of dimensional reduction, and do not claim a complete solution of the fully degenerate general problem.

Abstract: This paper presents a new way of defining convergence in analysis. We call it \emph{discordant convergence}. In this setting, the distance used to measure closeness between two functions is not fixed. Instead, it depends on the first function we are comparing. More precisely, we attach to each function $f$ a probability measure $\mu_f$. Then we define the distance from $g$ to $h$, using $f$ as the reference, as $\displaystyle D_f(g,h) = \int |{g-h}| \,d\mu_f$. A sequence $(u_n)$ converges to $u$ in the discordant sense if $D_{u_n}(u_n,u)$ goes to zero. This breaks with the standard idea that the metric is fixed once and for all. It allows comparisons that adapt to the objects being compared. We build a complete theory around this idea. First, we introduce discordant versions of Banach and Hilbert spaces. Then we prove that a concrete weighted $L^2$ example is complete. We also show that these spaces have natural forms of reflexivity and separability. The main technical result is a compactness theorem: every bounded sequence in the Sobolev space $W^{1,1}(0,1)$ has a discordantly convergent subsequence. Finally, we solve a nonlinear parabolic equation where the reaction term depends on the discordant distance. This example shows that the new framework can handle problems that are out of reach for classical methods. The results point to the broader relevance of the framework. It opens the door to a new class of adaptive, self-referential problems in analysis and its applications.

Abstract: Let Ω ⊂Rn be a bounded C2 domain and consider ut −∆u=∇uTA(x)∇u in Ω×(0,∞), u =0 on ∂Ω×(0,∞). We study only global finite strong solutions that attain the homogeneous Dirichlet condition as an ordinary trace and possess one Lipschitz time slice. For such a solution, if A ∈ L∞(Ω;Rn×n), then |u(x, t)| ≤ C0e−λ1(t−t0)ϕ1(x), t ≥t0, where (λ1,ϕ1) is the positive first Dirichlet eigenpair of −∆. The result is conditional on the existence of the stated global strong solution; no existence or uniqueness theorem is asserted. The proof uses the two transforms e±Ku, where K bounds the operator norm of the symmetric part of A, and therefore applies to bounded spatially dependent, anisotropic, and indefinite matrix fields. In the scalar isotropic quadratic case, decay at the heat-semigroup rate and the Cole–Hopf representation were established previously by Benachour, Dăbuleanu-Hapca, and Laurençot. We recall that formula only to compare the finite-trace regime with the profile −λ1t+logϕ1, whose boundary trace is −∞ rather than zero. Thus the present result addresses a finite-trace strong-solution subcase of Maz’ya’s Problem 55 and does not settle its unbounded or singular component.

Abstract: Problem 52 in Maz'ya's 2018 collection displays a capacitary lower bound for an unnormalized annular $L^p$-norm of a $p$-harmonic solution whose energy integral diverges at a boundary point. We prove that this displayed inequality, read literally, is false already for $p=2$ in every dimension $n\geq3$. In the upper half-ball, the upper-half-space Poisson kernel restricted to the domain, u(x)=xn|x|n, is positive and harmonic, has zero boundary values on the local flat boundary away from the origin, and has infinite Dirichlet energy. Its annular norm is computed exactly and satisfies ‖u‖L2(Bδ+∖Br+)∼Cnr−(n−2)/2, whereas the polynomial factor in the printed lower bound requires at least $r^{-(n-2)}$. The contradiction is independent of the capacitary exponential; for this domain the capacity term actually strengthens it. We also identify the normalization used in Maz'ya's linear theory: the growing-solution estimate is formulated through a fixed-ratio annular quadratic mean containing the factor $r^{-n}$, not through a raw $L^2$-norm. Accordingly, the counterexample invalidates the unnormalized 2018 display but does not settle the correctly normalized nonlinear Phragmén–Lindelöf problem.

Abstract: Let $n\ge 3$, let $\Omega\subset R^n$ be a bounded smooth domain, and put $Q=\Omega\times(0,T)$. We construct a finite nonnegative measure $\mu$, absolutely continuous with respect to Lebesgue measure and supported in a fixed compact subcylinder of $Q$, such that $$ \mu(K)\le C\,capheatQ(K) $$ for every compact $K\subset Q$, where $capheatQ$ is Pierre's classical heat-equation capacity. With the explicit half-closed-cylinder normalization introduced below, the same estimate holds for every compact $K\subset\Omega\times[0,T)$. Nevertheless, the first trace inequality in Maz'ya's Problem~56, $$ \int_Q |u|^2\,dd\mu \le C\left(\int_Q |\nabla_xu|^2\,dd x dd t + esssup_{0<t<T}\|u(t)\|_{L^2(\Omega)}^2\right), $$ fails even for $u\in C_c^\infty(Q)$. We state the exact $p=2$ variational capacity used in the proof, identify it with Pierre's capacity under the hypotheses of the construction, and distinguish the open-cylinder and half-closed-cylinder conventions. We also prove that a natural strong multilevel capacitary estimate would imply the trace inequality, and then exhibit nested compact sets and levels for which that estimate fails. Finally, we quantify why the same test sequence has divergent $L^2(0,T;H^{-1}(\Omega))$ time-derivative energy. Thus the one-set sufficiency suggested in Problem~56 is false for the first trace inequality in every dimension $n\ge3$. The second inequality in that problem and the cases $n=1,2$ are not resolved here.

Abstract: Maz'ya's Problem 43 asks whether the sharp planar condition for L^p-dissipativity of the Lamé operator remains sufficient in dimension three. We give a negative answer by an exact computer-assisted construction. At Poisson ratio \( \nu=0 \) and \( p=18+12\sqrt2 \), the proposed condition holds with equality. We exhibit an explicit nonvanishing periodic vector field and certify, through a finite Fourier dual certificate evaluated in exact rational arithmetic, that the associated transformed form has strictly negative mean. A quantitative localization argument converts this periodic obstruction into a compactly supported smooth test field in every nonempty open subset of \( R^3 \). The certificate, two verification scripts, a readable coefficient export, and cryptographic checksums accompany the source package.

Abstract: A linear bounded operator on a separable Hilbert space strict positive on a dense set is injective (Theorem 1, Par 2.). The result has been used as the backup for the criteria exploiting the operator approximation positivity properties on finite dimension subspaces having their union a dense set. The functional-numerical methods introduced are a consequence of the observation that, when the dense set is an infinite union of finite dimension subspaces from a family F, (S_n ∈ F, n ≥ 1), then the strict positivity on each subspace S_n of the operator approximations will attract the strict positivity on the dense set (Theorem 2) of the original operator provided that the positivity parameters of approximations are bounded by a strict positive constant. The criteria applied to the Alcantara-Bode integral operator connected to Riemann Zeta function, showed that its null space contains only the null element. That is in fact the equivalent formulation of the Riemann Hypothesis.

Abstract: Let $n>4$ and let $\capb$ denote the biharmonic capacity generated by the homogeneous $\dot H^2$ energy. Maz'ya asked whether the more general $m$-harmonic capacity is upper subadditive. We prove strict subadditivity for two externally tangent closed balls of arbitrary radii. The proof is based on inversion about the tangency point, which transforms the exterior of the union into an asymmetric slab. We define the clamped biharmonic Green kernels of the slab and the corresponding half-spaces by tangential Fourier transform and prove the capacity--Robin correspondence by a reverse Kelvin construction in the relevant homogeneous Sobolev class. The Robin comparison then reduces to the one-dimensional operator $(D^2-k^2)^2$. We derive the finite-interval diagonal Green function, establish the required low- and high-frequency bounds, and prove a strict inequality for every nonzero tangential Fourier mode. The final positivity step is obtained by an explicit coefficient formula for a two-variable entire function. Integration in the tangential frequencies yields the strict capacity inequality.

Abstract: We consider the planar Lane--Emden equation with a positive Robin parameter on a disk. The positive radial solutions are first parametrized by the logarithmic slope of a single normalized Lane--Emden profile; this yields exactly one positive radial solution for every Robin parameter. The zero-eigenvalue condition in the first angular sector is then reduced to the vanishing of an explicit scalar function \( F_p \). For every \( p\geq12 \), a phase--plane estimate proves that \( F_p \) is negative at a point where the logarithmic slope equals \( 1/2 \), whereas \( F_p \) is positive near both endpoints of its interval of definition. We select two sign-changing zeros and prove that both are simple. A mode-by-mode spectral analysis shows that, in a reflection-invariant space, the linearized kernel is one-dimensional at either zero and that the corresponding eigenvalue crosses transversally. The Crandall--Rabinowitz theorem therefore produces two local branches of positive nonradial solutions. Consequently, for \( p\geq12 \), uniqueness among all positive solutions fails for Robin parameters converging to two distinguished values, even though the positive radial solution remains unique for every parameter.

Abstract: Smart cities increasingly depend on chip-centered digital infrastructure, whose resilience is closely constrained by the upstream supply of semiconductor manufacturing equipment. Disruptions in lithography equipment trade may propagate downstream and threaten the security of urban digital systems. Existing studies on semiconductor trade networks have paid limited attention to the timing of long-term structural shifts and the mechanisms through which external shocks reshape network organization. Drawing on UN Comtrade data of lithography equipment from 2010 to 2024, this study employs complex network analysis to develop an integrated topological–spatial analytical framework. It combines break detection, event mapping, community evolution analysis, node role identification, and critical channel assessment. The results show that the global lithography equipment trade network has evolved into a structure characterized by core concentration, alliance-based community differentiation, and increasing dependence on critical cross-community channels. The findings suggest that, to enhance the resilience of chip-centered urban digital infrastructure, smart-city risk governance should: (1) be extended upstream to critical manufacturing equipment, (2) monitor high-leverage cross-community trade pathways as systemic-risk indicators, (3) prioritize alternative pathways and redundant configurations involving emerging brokerage nodes and upgrading core nodes. From the perspective of global high-end equipment, this study provides an empirical basis for assessing the resilience of smart-city digital infrastructure, particularly in identifying upstream critical dependencies, structural vulnerabilities, and potential systemic risks under external shocks.

Abstract: This paper is devoted to the construction and analysis of solution operators for a broad class of fractional-order systems. Both coupled systems with memory-decoupled structure and fully coupled systems are considered. Using Laplace transform techniques and matrix-valued operator methods, explicit representations of the associated operator families are derived. The developed framework extends classical fractional resolvent theory to distributed-order and fully coupled systems, highlighting the role of coupling in shaping the structure of solution operators. These operators provide a natural setting for the fractional Duhamel principle and thus play a central role in the analysis of nonhomogeneous problems. Finally, several examples are presented to illustrate the theory and demonstrate the explicit computation of solution operators for representative distributed-order and fully coupled systems.