Computer Science and Mathematics

Abstract: This paper introduces the radius of integrability, a quantitative invariant that transforms the qualitative ϵ-δ formulation of Riemann integration into a measurable property of function spaces. For a Riemann integrable function and a prescribed accuracy ϵ, the radius identifies the largest partition mesh δ that guarantees every tagged Riemann sum approximates the integral within the specified error. The framework is developed for both compact domain intervals, via pointwise and uniform radii, and unbounded intervals, through the tail integrability radius which quantifies the necessary truncation window for improper integrals. Key theoretical results include the establishment of a bottleneck identity relating local and global mesh requirements and a structural theorem showing that for C1 integrands, the radius is asymptotically governed by the inverse of the function’s total variation. Furthermore, this work completes a hierarchical program of regularity radii—encompassing convergence, continuity, and differentiability—by revealing a dimensional progression of geometric anchors. We demonstrate that while continuity is anchored at a point and differentiability at a line, integrability is anchored at a two-dimensional region. The theory is illustrated through explicit computations for several classical functions, including the normal density, the stretched exponential, and the Thomae function, providing a new quantitative lens for classifying integrable functions based on their partition sensitivity and tail decay regimes.

Abstract: This note extends our earlier existence and uniqueness results in AIMS Mathematics, 10 (2025), 19280–19299. Using sharper estimates for the squared Green’s functions and Rus’s fixed point theorem with the unweighted L2 metric, we obtain conditions for the classical problem with 1 < α ≤ 2 and the sequential problem with 1/4 < θ ≤ 1. A weighted L2 metric covers the full sequential range 0 < θ ≤ 1 and gives a less restrictive condition than the Banach condition based on the maximum L1 integral. We also compare the Rus and Banach conditions and illustrate the improvements with examples.

Abstract: Relator spaces provide a relational framework in which neighborhoods, interiors, closures, convergence and several topological properties can be generated directly from a family of binary relations. Independently, neutrosophic set theory extends classical and fuzzy descriptions by associating with every point three membership degrees representing truth, indeterminacy and falsity. In this paper we combine these two frameworks and develop a systematic theory of neutrosophic relator spaces. We define neutrosophic relator interiors and closures componentwise from a relator, and introduce neutrosophic open and closed sets, neutrosophic neighborhoods, induced neutrosophic topologies, continuity, separation properties, compactness and connectedness. Several basic algebraic and order properties of these operators are established. We show that the construction reduces to the corresponding theory of relator spaces when the three neutrosophic components are crisp, and to ordinary neutrosophic topological spaces when the relator is induced by a topology. Several examples are given, including finite relator spaces, symmetric relators, preorder relators and a crisp neutrosophic relator space. The resulting framework provides a common language for relational generalizations of neutrosophic topology and suggests further investigations involving convergence, quasiuniformity, bitopological structures and generalized separation axioms.

Abstract: For constant-coefficient higher-order linear difference equations on the nonnegative integers with values in a nonzero real or complex Banach space, the best Ulam constant is determined throughout the hyperbolic regime, where no characteristic root lies on the unit circle. Yuan, Xu and Brzdęk (2025) obtained a Green-function Ulam bound in this regime, but when there are at least two distinct characteristic roots and at least one lies inside the unit circle, neither the existence of a best constant nor the optimality of that bound was known. The Green bound is optimal for every hyperbolic root configuration. The key point is that bounded homogeneous corrections have no unstable component and hence decay at a uniform exponential rate. A finitely supported perturbation can then be translated far along the half-line and phase-aligned so that the corresponding Green correction asymptotically realizes the \( \ell^1\ \)-bound furnished by the Green kernel. Consequently, the best Ulam constant exists in every hyperbolic case and equals the \( \ell^1\ \)-norm of the Green kernel. The qualitative criterion that strong Hyers–Ulam stability is equivalent to the absence of characteristic roots on the unit circle was already known. Together with that criterion, the result gives the exact best constant whenever the equation is strongly Hyers–Ulam stable.

Abstract: We show that the Buzano–Dragomir inequality holds for octonionic Hilbert spaces. Using this generalization, we derive the Dragomir inequality for the octonionic numerical radius and an octonionic version of Deutsch entropic uncertainty principle.

Abstract: Let B \( \subset\ \)R² be the unit disk and let \( A_\beta \) be the positive Robin Laplacian associated with \( a_\beta(u,v)=\int_B \nabla u\cdot\nabla v\,dx+\beta\int_{\partial B}uv\,dS, \qquad \beta>0. \) For \(0<s\le 1\) we study the spectral fractional Lane–Emden problem \( A_\beta^s u=u^p,\qquad u>0\quad\hbox{in }B, \) where the Robin boundary interaction is encoded in the spectral definition of \( A_\beta^s \). The case \( s=1 \) is the classical planar Robin Lane–Emden equation. We first establish variational existence of positive solutions, and of positive radial solutions, in the fractional Sobolev-subcritical range \( p+1<2/(1-s) \). Positivity follows from the positivity-improving Robin heat semigroup and the semigroup formula for \(A_\beta^{-s}\). We then use the classical problem as a nondegenerate endpoint and develop a perturbation theory for \( s\uparrow1 \). In particular, the classical positive radial branch continues to a locally unique fractional radial sheet \( (s,\beta)\mapsto U_{s,\beta} \). The principal result concerns symmetry breaking. For each \( p\ge 8 \), let \(\beta_{p,-}\) and \(\beta_{p,+}\) denote the two distinguished simple first-angular symmetry-breaking parameters of the classical Robin problem established in the published local theory. We prove that, for \(s\) sufficiently close to \( 1 \), these two points persist as two \(C^1\) curves \(\beta_{p,\pm}(s)\) with \(\beta_{p,\pm}(s)\to\beta_{p,\pm}\) as \( s\uparrow1 \). At each point \( (s,\beta_{p,\pm}(s)) \), the fractional linearization has a two-dimensional first-angular kernel in the full space and a one-dimensional kernel in a reflection-invariant subspace. A self-adjoint conjugation of the linearized equation and spectral perturbation theory show that the corresponding simple eigenvalue of the bounded conjugated linearization crosses zero transversally. The Crandall–Rabinowitz theorem then yields local branches of positive nonradial solutions. Hence, for every \( p\ge8 \) and every fractional order sufficiently close to one, uniqueness among all positive solutions fails along parameters converging to each of the two fractional bifurcation curves, while the radial state remains locally unique within the radial class. We also give the extension formulation, the exact disk-scaling law, and explain why a genuinely variable-order model requires a different operator framework.

Abstract: We study the odd open spin-1 antiferromagnetic Heisenberg chain with positive endpoint fields through its reversible ground-state transform. The main results are two unconditional all-length obstructions to standard proof strategies. First, a high-magnetization projection recursion gives an explicit product bound on the fully polarized ground-state probability. For every positive field sequence \( h_N=o(N) \) this yields \( \alpha_N(h_N)\to 0 \), \( \qquad \limsup_{N\to\infty}\alpha_N(h_N)\log(N/h_N)\leq \frac{30}{7} \), for the ordinary logarithmic Sobolev constant of the ground-state process, with logarithmic tail coefficient at least (7/6). This does not imply collapse of either the Hamiltonian gap or the modified logarithmic Sobolev constant. Second, for the Diaconis–Stroock/Sinclair length-weighted congestion functional used in this paper, every routing with unit edge lengths has congestion at least linear in N; the same conclusion holds for uniformly comparable edge lengths, while routings forced through a polarized configuration obey a stronger rare-state obstruction. These are limitations of that comparison certificate, not upper bounds on the physical gap.As a complementary exact calculation, transfer matrices for the open AKLT ground space give both the singlet/triplet compression of \( W_m=\sum_{i=1}^{m-1}P_{1,i} \) and its coupling to the orthogonal complement. The compressed spectral diameter is \( O(m3^{-m}) \), whereas \( ||Q_mW_m\mathsf P_m||\sim\sqrt{2(m-1)}/3 \). This finite-block statement quantifies why the bare compressed splitting alone does not control the perturbation at the Heisenberg point; it is not a replacement for a locality-sensitive AKLT stability theorem. Exact rational inertia and interval arithmetic then provide supporting finite-volume certificates: full gaps and the complete AKLT–Heisenberg interpolation are certified for \( N=3,5 \), and at \( N=5,h=1 \) one obtains both a negative pointwise Bakry–Émery curvature witness and [ 27/25<\Delta_5(1)<567/500. \( 27/25<\Delta_5(1)<567/500 \)These finite-volume statements have no asserted local-to-global consequence. The fixed-field uniform boundary-gap problem remains open.

Abstract: Gradient clipping is widely used to control large updates in optimization, while finite-step gradient descent develops period-two oscillations near the edge of linear stability. Recent work has analysed these oscillations via smooth flip bifurcation theory and higher-order stability criteria, whereas piecewise-smooth bifurcation theory describes generic interactions between period-doubling and switching boundaries. It remains unclear how a smooth gradient-descent oscillation first interacts with the switching surface generated endogenously by radial gradient clipping, and what dynamical structure emerges after this contact. For the finite-dimensional clipped gradient map \(F_{\eta,c}(x) = x - \eta \, \mathcal{C}_c^\diamond(\nabla V(x))\), we analyse this transition near a nondegenerate strict local minimum with a simple maximal Hessian eigenvalue. Clipping is locally inactive near the minimum, so the fixed-point stability threshold and the local flip bifurcation coincide with those of ordinary gradient descent. Under a supercritical flip condition, the stable period-two branch reaches the clipping boundary at \[ \eta_{\mathrm{sc}}(c) = \eta_f + K_\diamond c^2 + O(c^4) \] with \(\eta_f = 2/\lambda_*\), where \[ K_\diamond = \Gamma_*/(3\lambda_*^4 \|v_*\|_\diamond^2) \] is determined explicitly by derivatives of the potential up to fourth order and by the clipping norm. Beyond the separation law, the gradient structure yields rigidity absent from generic switching maps: the two points of every smooth period-two orbit have opposite gradients and hence reach a radial clipping boundary simultaneously; mixed smooth--clipped period-two itineraries are impossible; and every fully clipped period-two orbit has an exact unit Floquet multiplier and is therefore nonhyperbolic. A solvable planar model realizes the full transition from a stable smooth two-cycle to a neutral family of fully clipped period-two cycles.

Abstract: We show that for certain classes of matrices over octonions, a weak version of breakthrough numerical radius inequalities obtained by Kittaneh [Stud. Math., 2005] hold.

Abstract: Dragomir \( [\textit{Bull. Aust. Math. Soc., 2016}] \) showed that the Buzano inequality holds for orthogonal projections on Hilbert spaces. We show that Dragomir result extends to adjointable morphisms on Hilbert C*-modules. Dragomir \( [\textit{Linear Multilinear Algebra, 2016}] \) also derived the most general form of the Buzano inequality for bounded linear operators on Hilbert spaces. We show that Dragomir result extends to adjointable morphisms on Hilbert C*-modules. Using this generalization, we derive bounds for the roots of polynomials over commutative unital C*-algebras. We formulate the notion of noncommutative numerical range and derive numerical radius bounds for self-adjoint morphisms on Hilbert C*-modules over unital C*-algebras. We formulate several open problems, including the noncommutative Toeplitz-Hausdorff, von Neumann inequality, Ando inequality, Berger power dilation, Kittaneh inequality, Crouzeix problems.

Abstract: For the planar Lane--Emden equation with a positive Robin parameter, an earlier result gives two local symmetry-breaking points for \( p\ge8 \). We establish a global classification of all first-angular degeneracies along the positive radial branch. A unique exponent \( p_*\in(6.39,6.40) \) separates full linearized nondegeneracy from exactly two simple first-mode degeneracies. At \( p_* \) the first angular eigenvalue has a quadratic contact with zero; above it the two Robin parameters split at square-root rate and each gives a local branch of positive nonradial solutions. A rational interval certificate proves the numerical bracket.

Abstract: We develop a unified framework for anisotropic mixed fractional inequalities in which the fractional order and the integrability exponent may vary from one coordinate direction to another. Within this setting we establish a general family of Gagliardo--Nirenberg inequalities for the anisotropic mixed fractional Sobolev spaces, with an explicit homogeneity condition that corrects earlier formulations and with constants that depend on the effective dimension rather than on the ambient dimension. In the critical regime where the product of the fractional order and the integrability exponent equals the homogeneous dimension, the Sobolev embedding into the space of essentially bounded functions fails; we prove sharp logarithmic corrections that compensate for this failure, and we show by means of explicit extremal sequences that the logarithmic factor is optimal. We further develop a weighted theory for Muckenhoupt weights, extending the maximal, Gagliardo--Nirenberg, and Landau inequalities to weighted anisotropic mixed fractional Sobolev spaces with constants that depend polynomially on the Muckenhoupt characteristic. Optimal constants and extremal functions are studied through a variational problem whose Euler--Lagrange equation is a fractional Poisson-type equation with measure-valued source, and existence of extremals is proved by concentration-compactness. As applications, we derive a priori gradient bounds and sharp blow-up criteria for anisotropic fractional parabolic equations, and we prove that anisotropic neural networks achieve the optimal approximation rate governed by the effective dimension, with matching Kolmogorov width lower bounds. The resulting theory provides a common analytical framework linking fractional calculus, anisotropic function spaces, partial differential equations, and approximation theory. The endpoint cases of integrability are left as open problems.

Abstract: Let \(\Omega\subset\mathbb R^N\) be a bounded \(C^{2,\alpha}\) domain. We first refine the critical quantitative Alexandrov estimate at \(p=(N-1)/2\). Writing \(H\) for the normalized mean curvature, \(H_0=|\partial\Omega|/(N|\Omega|)\), and \(\varepsilon_H=\|H-H_0\|_{L^p(\partial\Omega)}\), we prove an annular estimate of order \(\varepsilon_H[1+\log^+(C_*/\varepsilon_H)]^{(N-3)/(N-1)}\), uniformly on bounded \(C^{2,\alpha}\) geometric classes. This improves the available full-logarithmic critical profile, and logarithmically concentrated perturbations of the sphere show that the exponent \((N-3)/(N-1)\) is optimal. We then apply the geometric estimate to the Dirichlet heat equation with initial temperature 1. A uniform short-time expansion yields \(\|H-H_0\|_{L^r}\le C[\delta_r(t)+t^{\alpha/2}]\), where \(\delta_r(t)\) is the \(L^r\) distance of the boundary heat flux from the constants. Combining this transfer principle with quantitative Alexandrov stability gives a complete sharp heat-flux stability profile: linear in the supercritical regime, fractional-logarithmic at the critical exponent, and power-like in the subcritical regime. In particular, asymptotic \(L^r\) constancy of the short-time boundary heat flux forces \(\Omega\) to be a ball.

Abstract: We introduce a family of polynomial vector fields on the plane generated by factorization in the ring of radial invariants. This family, referred to as the Radial Factorization Family, is generated by coupling an arbitrary monic polynomial \( P \) in the radial invariant \( \Delta=x^{2}+y^{2} \) with the standard rotational generator. It admits an explicit reduction in polar coordinates, yielding a clean global unperturbed model whose periodic orbits are concentric circles with explicitly known radii. By adding a small polynomial perturbation that breaks rotational symmetry, we compute the first-order Melnikov function and establish that the displacement function of the Poincaré return map admits the expansion \( d(r,\varepsilon)=2\pi\varepsilon\,\mathcal{M}(r)+O(\varepsilon^{2}) \) on compact subintervals of the period annulus. Because the unperturbed orbits are circles, the resulting integrals admit explicit evaluation. For the linear center this recovers the classical bound \( \lfloor(d-1)/2\rfloor \) on the number of bifurcating limit cycles; near a hyperbolic invariant circle of a general member of the family the same framework shows unique persistence of a single limit cycle, while in the degenerate case of multiple roots the circles lose normal hyperbolicity and unfold under perturbation into at most \( m \) limit cycles controlled by the multiplicity. The central novelty of the construction lies in its inverse character: in higher dimensions it yields an optimal inverse-realization procedure that converts any prescribed collection of nested invariant spheres—together with a topologically admissible assignment of stability types—into a polynomial vector field of minimal degree within the radial-factorized equivariant class. This provides a systematic algebraic dictionary for topological synthesis, transforming the traditional paradigm of "analyse a given system" into "t design a system with prescribed dynamical skeleton". The framework provides a constructive algebraic perspective related to aspects of Hilbert's sixteenth problem and offers a practical tool for phase-space engineering in nonlinear dynamics.

Abstract: Let \( A \) be an invertible normal hyperbolic operator on a complex Hilbert space \( H \). We obtain an exact spectral formula for the optimal uniform shadowability constant \( Shad(A) \), sharpening the general additive Green-operator estimates to a Euclidean combination of the stable and unstable spectral contributions. If both spectral components are nonempty, set \( \alpha(A)=1-\max\{|\lambda|:\lambda\in\sigma(A),\ |\lambda|<1\}, \) and \( \beta(A)=\min\{|\lambda|:\lambda\in\sigma(A),\ |\lambda|>1\}-1. \) We prove the exact formula \( Shad(A)=\left(\alpha(A)^{-2}+\beta(A)^{-2}\right)^{1/2}, \) with the corresponding one-sided formulas when only one spectral component is present. We further relate this exact constant to the robustness of hyperbolicity. Let \( r_{\mathrm{hyp}}(A) \) denote the operator-norm distance from \( A \) to the loss of hyperbolicity. We obtain the sharp universal ratio \( 1\le r_{\mathrm{hyp}}(A)Shad(A)\le\sqrt{2}. \) Both constants are optimal. In the mixed case, \( r_{\mathrm{hyp}}(A)Shad(A)=\left[1+\left(\frac{\min\{\alpha(A),\beta(A)\}}{\max\{\alpha(A),\beta(A)\}}\right)^2 \right]^{1/2}. \) Thus the normalized shadowability constant is determined exactly by the relative balance of the stable and unstable spectral gaps.

Abstract: Consider ratios of adjacent normalized Maclaurin remainders after perturbing a finite block of consecutive coefficients of \(\tan^2x\). For \(r\) perturbed coefficients, a general power-series monotonicity theorem gives at most \(r\) interior critical points. For the tangent-square coefficients, this bound is attained for every \(r\) by an explicit order-\(r\) degeneracy with a full-rank local unfolding. The one-coefficient case has an exact classification, including sharp ranges, monotone motion of the maximizer, and parameter and order asymptotics. Two consecutive perturbations admit a two-parameter critical-point phase diagram and an exact classification of the global extrema. Unique switch functions determine when the interior maximum and minimum overtake the relevant endpoints. The upper bound follows from classical variation diminishing principles. Sharpness and the two-parameter phase diagram use the special structure of the tangent-square coefficients.

Abstract: For \( \operatorname{arsinh}(x+\lambda x^3+\eta x^5) \), consider the best cubic bounds with linear term \(x\). When \(\eta=0\), a complete classification of the normalized remainder determines the optimal coefficients on \([0,1]\) for every \(\lambda\geq0\). For positive \(\lambda,\eta\), a parameter-uniform sign-variation argument gives at most two positive critical points. For fixed \(\eta\), the critical-point regimes occur in a fixed order as \(\lambda\) increases. The fold has a unique endpoint height \(\eta_c\). A positive fold threshold exists exactly for \(\eta_*<\eta<\eta_c\). The fold terminates transversely at \(\lambda=0\). Near \((\lambda_*,\eta_*)\), two analytic curves mark the appearance of an interior minimum and its equality with the boundary value. Across the second curve, the minimizing point changes discontinuously and the optimal lower coefficient is continuous but not differentiable. The local sixth-order pattern is classical. The critical-point classification is global for the full composite function. Exact sign bounds also give a rational example.

Abstract: For a positive-coefficient polynomial p(x) = 1 + x + Σ_{j=2}^m a_j x^j, set Q_p(x) = (log p(x) - x)/x^2. Then Q_p has at most m - 1 positive critical points, and this bound is attained for every m ≥ 3. An explicit positive truncation determined by the largest zero of a probabilists’ Hermite polynomial has an order-(m - 1) critical degeneracy at the origin. At this reference polynomial, the coefficient-to-jet map has full rank, so nearby positive-coefficient polynomials realize prescribed ordered configurations of m - 1 sufficiently small positive simple critical points and hence attain the bound. For the cubic polynomial 1 + x + λx^2 + ηx^3, the positive critical points are classified by a two-parameter phase diagram whose fold has a unique endpoint on λ = 0. A separate local curve marks the switch in the sharp lower quadratic coefficient. The degree-two case admits exact sharp quadratic and cubic coefficient bounds together with all parameter transitions.

Abstract: We study normalized integrals of the Widder expressions associated with \(F_\xi = 2G'/G\), where \(G(x) = \xi(1/2 + \sqrt{x})\). Their zero weights are regularized incomplete beta kernels. We prove a two-term zero-count formula with error \(O_L(\log x)\), uniformly for integers \(1 \le m \le Lx\) and each fixed \(L > 0\). The proof controls the effect of replacing complex reflected zeros by their ordinates at the same uniform scale. As a consequence, these integrated expressions recover \(N(\sqrt{x})\) to error \(O_{a,L}(\log x)\) when \(a\sqrt{x} \le m \le Lx\). Every cumulative lower density of simple critical-line zeros transfers to the corresponding smoothed sums; Lamzouri's stated proportion theorem supplies the value \(0.6725007\ldots\). Separately, we apply the finite Stieltjes framework of Bondesson--Simon to verified zero information, obtaining an explicit range of global Widder positivity. This application neither improves the verified height nor implies the Riemann hypothesis.

Abstract: The information technology (IT) sector is one of the fastest growing contributors to employment and export earnings in Pakistan, yet timely and detailed evidence on the skills and roles employers actually demand remains scarce. This study addresses that gap by analysing 3,734 IT job advertisements collected from LinkedIn for locations in Pakistan. The advertisements span 1,794 distinct job titles, 1,420 companies, and 49 cities. Postings were manually collected from publicly visible listings, cleaned, and classified into six IT domains and twenty-two subcategories, and were examined along the dimensions of occupational structure, geographic distribution, employer concentration, work arrangement, employment type, industry, and applicant competition. To compare competition across advertisements of differing ages, an exposure-adjusted measure of application intensity (applicants per posting-hour) was used. The results show that Software Development dominates advertised demand (75.76% of postings), while Artificial Intelligence and Data Science together form a substantial and highly contested second cluster (20.17%). Competition is strongly inverse to volume: specialised fields such as Computer Vision (9.18) and Data Science (8.93) attract far higher application intensity than high-volume categories such as Software Development (1.71). Hiring is geographically concentrated in Punjab, Sindh, and the Islamabad Capital Territory, and is dominated by a small number of large employers. Remote and full-time positions attract disproportionately high competition. The findings provide current, platform-specific evidence on the structure of Pakistan’s advertised IT labour market and demonstrate a transparent, reproducible framework for deriving labour-market intelligence from public job postings, while explicitly acknowledging the interpretive limits of single-platform vacancy data.