Computer Science and Mathematics

Abstract: Sensitivity analysis plays a significant role in optimization models arising in applied sciences such as financial mathematics, risk analysis, signal processing, neural networks, and optimal control. In order to analyze the stability and qualitative behavior of solutions, parametrization of objective functions and constraints is frequently employed. While convex optimization problems ensure existence and stability of global optima under mild assumptions, non-convex optimization problems pose substantial theoretical challenges due to the possible absence of global optimal solutions. In this paper, we investigate sensitivity analysis for a class of generalized parametric non-convex optimization problems, where both the objective function and constraints may be non-convex. By introducing small perturbations through parameters, we analyze the behavior of local optimal solutions and derive results on the subdifferential of the optimal value function. Our approach extends several known results from convex parametric optimization to broader classes of generalized non-convex functions, including pseudo-convex and invex functions. We establish theoretical results with detailed proofs and validate our findings through numerical examples.

Abstract: Mathematical notation is standardly treated as a recording convention. This paper argues that notation is a compression technology and that the elementary arithmetic hierarchy—counting, addition, multiplication, exponentiation—is a compression hierarchy: each level compresses iterated application of the level below, and each act of compression renders cheaply expressible a class of structure that was expressible below only at prohibitive description-length cost. Primes are, in this sense, invisible from inside addition; scaling laws are invisible from inside the additive representation of magnitude. Two principles organize the argument. First, successful compression is a measurement of structure: compression is possible only where regularity exists and fails, informatively, where it does not. Second, a compressed representation converts the exploited regularity into a manipulable object governed by the arithmetic of the level below—multiplication of numbers becomes addition of exponents—so that each level is a cognitive instrument rather than a shorthand. A two-part criterion distinguishes notational compression from algorithmic improvement: the replaced representation prices all objects of its domain uniformly, and the compressed parameter acquires laws of its own. Instances of the pattern in mechanics, quantum state representation, and neural networks are outlined and reserved for separate treatment.

Abstract: This study investigates gender differences in mathematics anxiety among young adults in Bangladesh and examines how educational background, social perceptions, and gender norms may contribute to this phenomenon. A comprehensive multiple-choice survey was conducted among 227 individuals aged 18–25 to assess mathematics anxiety and identify factors associated with students' experiences of mathematics. The participants represented two major educational pathways in Bangladesh: the National Curriculum and Textbook Board (NCTB) curriculum, and the British Curriculum. The findings reveal that female students experience significantly higher levels of mathematics anxiety than male students. The results also indicate differences in perceptions of mathematical proficiency across educational backgrounds, suggesting that students' experiences may vary according to both gender and curriculum. Mathematics anxiety can negatively affect academic performance, mental well-being, motivation, and willingness to pursue mathematics-intensive subjects and careers. Notably, many respondents perceived gender as a barrier to accessing mathematics- and STEM-related career opportunities in Bangladesh, highlighting a potential connection between mathematics anxiety and the underrepresentation of women in mathematics-based professions. To quantify individual anxiety severity, we developed a Python-based computational algorithm incorporating multiple factors from the survey responses. This approach enabled the identification of demographic patterns that may be overlooked through conventional analysis.

Abstract: We study in the setting of an arbitrary metric space (X, d) hyperbolic-type metrics that are generalizations of two stabilizing metrics introduced and studied by Boskoff and Suceavă, induced by a version of Barbilian’s logarithmic oscillation. These metrics are defined on X∖M, where M⊂X is a nonempty proper subset, using a function F:X∖M→(0,∞) as a counterpart of the distance to M and a function f:[0,∞)→[0,∞) that vanishes only at the origin, is nondecreasing and subadditive. The first stabilizing distance between x,y∈X∖M is defined as log(max{F(x),F(y)}/min{F(x),F(y)}+f(d(x,y))). The second stabilizing distance between x,y∈X∖M is defined as log((max{F(x),F(y)}+f(d(x,y)))/min{F(x),F(y)}). For these two generalized stabilizing metrics and for a generalized version of Vuorinen’s distance ratio metric, using natural assumptions on F and f, we provide sufficient conditions for the Gromov hyperbolicity and we show that the metric space (X∖M,ρ) is complete provided that (X,d) is complete, then we find an upper bound for the linear dilatation of the identity map 1_X∖M:(X∖M,d)→(X∖M,ρ). The main result of the paper shows that, given a Ptolemaic metric space (X,d), the first generalized stabilizing metric on X∖M, with f=1[_0,∞) the identity map of [0,∞), is log2−hyperbolic in the sense of Gromov.

Abstract: For a complex matrix \(A\), let \(\psi(A)\) be the norm of the polynomial functional calculus on its numerical range. We prove that \(\psi(A)=2\) only when the numerical range is a nondegenerate closed disk. The proof first extracts the equality conditions in the sharp double-layer estimate. After boundary eigenvalues are removed, an extremizer obtained from a finite Blaschke product satisfies an exact boundary-kernel identity. On a curved exposed arc this identity makes the extremizer rational. A direct resolvent argument then places every zero of its reduced numerator inside the numerical range and every pole outside; consequently its full unit lemniscate coincides with the boundary of the numerical range. The defining polynomial of the projective dual of that algebraic boundary divides the Hermitian Kippenhahn determinant. Hyperbolicity and strict convexity force the dual curve to have degree two, and its points at infinity force the resulting conic to be a circle.

Abstract: We introduce a double golden-ratio Tseng-type extragradient method (DGR--Tseng) for solving variational inequality problems in real Hilbert spaces. The method exploits two golden-ratio averaging steps while requiring only a single projection per iteration. Weak convergence is established under a solution-oriented condition weaker than monotonicity and pseudomonotonicity, without imposing any finiteness assumption on the solution set. Under a Robinson/Luo--Tseng-type local error-bound condition, we further establish \(R\)-linear convergence of the distance to the solution set and a \(Q\)-linear contraction of an associated weighted paired distance, without requiring strong monotonicity, strong pseudomonotonicity, or a singleton solution set. The error bound is weaker than the strong monotonicity-type assumptions commonly used to obtain linear convergence and does not require the solution set to be a singleton. The stepsize is updated adaptively, eliminating the need for prior knowledge of the operator's Lipschitz constant. Numerical experiments on sequence-space and function-space variational inequality problems, together with applications to sparse signal reconstruction and multi-OD urban traffic network equilibrium, demonstrate the computational effectiveness of DGR--Tseng compared with several single golden-ratio methods.

Abstract: In this paper, we introduce an adaptive relaxed two-inertial Tseng method for solving a class of variational inclusions in real Hilbert spaces without assuming monotonicity of the composite operator. The method combines two inertial displacements, a Tseng forward correction, and a relaxation step. Its adaptive stepsize does not require prior knowledge of the Lipschitz constant of the single-valued operator. Weak convergence is established under a mild condition that compares graph points only with zeros of the composite operator, together with an algorithmic cluster-identification condition. For the rate analysis, we avoid the commonly imposed assumptions of strong monotonicity and strong pseudomonotonicity. Instead, R-linear and Q-linear convergence are established under a less restrictive error-bound condition that does not require the solution set to be a singleton. Numerical experiments involving two finite-dimensional nonmonotone inclusions and three color-image restoration problems show that, under the reported settings, the proposed method requires fewer iterations and less CPU time than the three comparison methods.

Abstract: This study examines dynamic volatility transmission among WTI crude oil, gold, the U.S. Dollar Index (DXY), and the Stock Exchange of Thailand (SET) using a Bayesian TVP-VAR-SV framework. Using 4,206 daily observations from 2008 to 2025, the study combines generalized forecast-error variance decomposition, dynamic connectedness, directional measures, generalized impulse responses, and crisis-regime analysis. Volatility connectedness intensifies markedly during financial stress, with the Global Financial Crisis (GFC) recording the highest mean TCI (3.0522), compared with 1.6598 during Normal periods. Gold emerges as the dominant net transmitter during the GFC, while the USD becomes the strongest receiver, with Gold-to-Oil, Gold-to-USD, and Gold-to-SET transmission also increasing substantially. GIRF results further show that Gold shocks generate immediate responses across markets but attenuate rapidly, indicating strong yet short-lived transmission. Importantly, market roles are state dependent: during the GFC, gold becomes a major transmitter within a financial-system crisis, whereas during the Russia–Ukraine episode, gold becomes a receiver and the USD a transmitter amid geopolitical and real-economy shocks. These findings highlight that both the direction and persistence of volatility transmission depend on the nature of the underlying shock.

Abstract: We give two proofs of the finite-dimensional scalar Crouzeix theorem: the numerical range of every complex matrix is a 2-spectral set for polynomials and for rational functions without poles on the numerical range. Both proofs start from the positive double-layer calculus and reduce the sharp local estimate to a top singular pair, but their certificates are different. The first disintegrates the boundary density over the fibres of a finite Blaschke product, uses conjugate Cauchy companions and Fourier cancellation, and forces an impossible unbounded scalar recurrence. The second keeps the boundary variable, constructs a positive defect-Gram family, and obtains a weighted telescoping identity whose strictly positive first defect excludes norm at or above 2. We identify the common scalar moment sequence underlying the two certificates. A shared finite confluent Schur completion, real-analytic convex outer exhaustion, and intrinsic rational calculus give the global theorem. Two pinned Lean 4 developments separately kernel-check the corresponding polynomial and reduced-rational endpoints.

Abstract: This article investigates a class of nonsmooth interval-valued multiobjective programming problems with vanishing constraints (NIMPPVCs) on Hadamard manifolds, under locally Lipschitz continuity assumptions on objective and constraint functions. We establish that the various standard constraint qualifications, namely Cottle-type, Slater-type, Mangasarian-Fromovitz-type, and linearly independent constraint qualifications, usually violate at any feasible point of NIMPPVC. Moreover, we introduce several NIMPPVC-tailored constraint qualifications for NIMPPVC, in particular, Abadie constraint qualification (ACQ-VC), generalized Abadie constraint qualification (GACQ-VC), generalized Guignard constraint qualification (GGCQ-VC), Cottle-type constraint qualification (CCQ-VC), Slater-type constraint qualification (SCQ-VC), Mangasarian-Fromovitz-type constraint qualification (MFCQ-VC), and linearly independent constraint qualification (LICQ-VC), and further establish interrelations among them. In addition to this, by employing GGCQ-VC, we establish the Karush-Kuhn-Tucker (KKT)-type necessary optimality conditions for LR-efficient solutions of NIMPPVC via Clarke subdifferentials. Moreover, sufficient criteria of optimality for NIMPPVC are derived under generalized geodesic convexity hypotheses and certain mild restrictions on the index sets. Furthermore, the sufficient optimality conditions established in this paper are applied to propose an algorithm for identifying the LR-efficient solutions of NIMPPVC. Various illustrative examples on Hadamard manifolds are furnished to highlight the significance of the results derived in this paper. To the best of our knowledge, constraint qualifications and optimality conditions for NIMPPVC have been investigated for the first time in the Hadamard manifold framework.

Abstract: We introduce a deterministic iterative transformation T on four-digit strings. The transforma- tion places the four digits on the vertices of a square, replaces each edge by the digital root of the sum of its two endpoint digits, and subtracts the ascending arrangement of the four re- sulting edge-values from the descending arrangement. We call this the KHOKHAR Square Digital Root Transformation. An exhaustive computer search over all 104 four-digit strings from 0000 to 9999 shows that T has exactly two fixed points, 0000 and 7443, and that every four-digit string reaches one of them within at most six iterations. We give a short, direct proof characterizing exactly which strings map to 0000: a string does so if and only if its four edge digital roots already coincide. This condition holds for exactly 136 of the 10,000 four-digit strings , the ten repdigits together with 126 further, non-repdigit strings such as 0101 and 9990 , which refines the natural first guess that repdigits alone are responsible. The remaining 9,864 strings all converge to the nontrivial fixed point 7443. We close by discussing applications of the construction, and of its exhaustive-verification proof method, in computer science and quantum computing.

Abstract: Class ordinals are first defined as an extension of ordinals, which are called set ordinals. In order to create a satisfactory definition of a maximally large class, this definition needs to quantify over all set-theoretic theories. Consequently, the Axiom of Maximal Strength (AMS) asserts that there exists a class that does not provably exist in any theory that has a theorem-preserving interpretation into a consistent theory that has a parameterless first-order definition with the length of a set ordinal. This property is called non-SO-interpretability. The meta-theory T_M that interprets AMS is presupposed to be sound, non-SO-interpretable, and to extend MK. It is then proven that T_M is consistent with AMS and that the class Ord of all the ordinals of T_M is a consistent version of Cantor's absolute infinite.

Abstract: Let \(\Phi\) be the theta kernel of the Riemann \( \Xi \)-function and \( D(z)=\int_0^\infty\Phi(u)\cos(zu)\,du \). The Riemann hypothesis is equivalent to the positivity of \( \partial_y|D(x+iy)|^2$ for $y>0 \), and it is natural to seek that positivity by decomposing the growth derivative over the phase-aligned blocks \( J_m=[m\pi/x,(m+1)\pi/x] \) on which the oscillation completes a full period, and controlling each block separately. We derive the exact two-variable representation of \( \partial_y|D|^2 \), obtain the resulting longitudinal--transverse decomposition \( \partial_y |D|^2 = 2 \int_{0}^{\infty} \big[ Q_y(a)\sin(2xa)+\varepsilon_x(a;y) \big] da \), and establish a complete oscillatory hierarchy: the structural fact is that a phase-aligned block annihilates one Taylor order against a sine but two against a cosine. On compact phase regions where \( Q_y' \) is bounded away from zero, the longitudinal block is of order \( x^{-2} \) while the transverse residual is two powers smaller, with explicit leading coefficients. We then prove that the decomposition cannot localise the positivity. Writing \( S=\sum_mC^{(Q)}_m \) and \( E=\sum_{m}C_{m}^{(\varepsilon)} \), both sectors admit complete algebraic asymptotic expansions in \( 1/x \), and these are termwise negatives of one another: \( S+E \) is exponentially small while each sector is of size \( x^{-5} \), so that \( E/S\to-1 \)(Theorem 5). As a concrete consequence we obtain a representation-specific no-go theorem for universal phase-aligned block positivity: for every \( y>0 \) and all sufficiently large \( x \) at least one aligned block is negative (Theorem 6). The latter proof is fully analytic and unconditional, resting only on the expansion\( Q_y(a)=\frac{4}{3}y\,p'(0)\Phi(0)^2a^4+O(a^6) \) together with a rigorous proof that \( \Phi''(0)<0 \); in particular it is independent of the truth of RH. Within this exact phase-aligned decomposition, therefore, the strategy of deducing global positivity from positivity, or from independent domination, of every block is not merely unproved but impossible: positivity is intrinsically nonlocal.

Abstract: For a Hutchinson iterated function system (IFS), a Banach contraction on a complete metric space, or a finite-metric dynamical system, a natural question is: at which resolution σ does the contraction's geometric structure (fractal attractor, basin of attraction, periodic part) become optimally visible? We answer this by introducing a scale-selection principle: define the observation scale σ_c := argmax_σ χ(σ) where χ(σ) = |dO(σ)/dlogσ| is the susceptibility of a resolution-dependent observable, and prove that σ_c exists under explicit boundary-regularity hypotheses.The framework's main quantitative results are three theorems specialised to Banach contractions and IFS: (i) a geometric scaling identity σ_c = qL for affine Banach contractions with operator norm q and basin scale L, applying to Hutchinson IFS with σ_c ∼ q · diam(K_⋆) for the same observable; (ii) a discrete Banach theorem on finite metric structures under uniform Lipschitz Lip_d(f) = q < 1, giving an exact collapse-time N^⋆ = ⌈log(∆/d_min)/log(1/q)⌉; (iii) a spectral concentration theorem placing σ_c at the inverse log-spectral-gap of the transfer operator at fixed positive noise. A stability lemma for canonical normalisation under smooth windowing and a parametric Banach correspondence observation complete the technical core.The framework is stated explicitly in the non-expansive Lipschitz regime Lip_d(f) ≤ 1 on the metric side, and at fixed positive noise ε ∈ (0, 1) on the spectral side. A four-type classification of operations by injectivity structure organises the broader landscape; cross-domain empirical evidence anchored on a peer-reviewed NISQ-hardware measurement of σ_c is summarised. The middle-thirds Cantor set IFS appears as the principal worked example.

Abstract: We introduce a global approach to uniqueness questions for real Z-eigenvectors of even-order symmet-ric tensors. Motivated by the Lusternik–Schnirelmann (LS) category lower bound, we call an order-2k symmetric tensor in dimension n ≥ 2 eigenminimal if it has exactly n real projective Z-eigendirections. We construct three explicit mechanisms producing eigenminimal tensors and show that, in the stated parameter ranges, the resulting tensors have at most one Z-eigenvector in the strict positive cone. We also exhibit an eigenminimal binary quartic having two projective Z-eigendirections that meet the strict positive cone, showing that positive uniqueness is not a consequence of eigenminimality alone. The three model families, in the stated parameter ranges, lie in a single path component of the eigenminimal locus. Furthermore, nondegenerate eigenminimality is stable under perturbation, so the eigenminimal locus has nonempty interior but, for orders at least four, is not dense.

Abstract: For an \( n\times n \) complex matrix \( A \), let \( W(A)=\{x^*Ax : x\in\mathbb{C}^n,\ \lVert x\rVert_2=1\} \) be its numerical range. Here \( x^* \) denotes the conjugate transpose, \( \lVert\,\cdot\,\rVert_2 \) the Euclidean norm, and \( \lVert\,\cdot\,\rVert \) the induced operator norm. For every complex polynomial \( p \), we prove that \( ‖p(A)‖≤2maxz∈W(A)|p(z)| \). More generally, the same estimate holds for every function \( f \) holomorphic on a neighborhood of \( W(A) \), with \( p \) replaced by \( f \); consequently, \( W(A) \) is a 2-spectral set for \( A \). This proves Crouzeix's conjecture, and the constant is optimal. The central ingredient is a mass-parameterized positive-real completion theorem relative to an auxiliary eigenbasis: if a normalized matrix-valued Carathéodory function completes \( \frac{2}{\mathfrak m}(I-wT)^{-1} \) modulo the adjoint algebra, then \( \lVert T\rVert\leq\mathfrak m \) for \( \mathfrak m\geq2 \). The numerical-range double layer has mass two. The classical positive double-layer calculus supplies such completions for \( f(B) \) whenever the auxiliary matrix \( B \) has simple spectrum; the eigenvalues of \( f(B) \) may repeat. Sampling the associated Herglotz kernel at scaled conjugate diagonal entries and at the origin cancels the nonconstant completion term and leaves two ordered weighted Gramians; their first nonconstant terms yield the mass bound. Simple-spectrum approximation and canonical convex outer domains with supported oriented radial boundaries yield the general theorem. The mass formulation also provides a concrete scalar proof protocol for annular, multiply connected, operator-radius, and abstract spectral-constant problems. For a complex square matrix A, we prove the sharp polynomial Crouzeix inequality and its extension to every function f holomorphic near the numerical range of A. Consequently, the numerical range is a 2-spectral set, and the factor 2 is optimal. The proof uses a mass-parameterized positive-real completion theorem relative to an auxiliary eigenbasis. A matrix-valued Herglotz kernel is sampled so that the adjoint-algebra correction cancels, leaving ordered weighted Gramians that yield the sharp mass bound. Simple-spectrum approximation and convex outer domains give the general result. The mass formulation also supplies a reusable scalar protocol for sharp annular, operator-radius, and abstract spectral-constant problems.

Abstract: Let f(x) be a monic integral polynomial of degree n and p a prime number for which f(x) is fully decomposable modulo p. Let integers r₁,...,r_n be the roots of f(x) mod p with 0 ≤ r₁ ≤ ··· ≤ r_n < p. Inthis series of papers, we have investigated the distribution of the points (r₁,. . .,r_n). In the present paper, we present several conjectures concerning the distribution of polynomials in the roots r_i.

Abstract: This article deals with so-called finite-rank solutions, originally introduced by Laplace for linear second-order partial differential equations (PDEs) in the plane. They consist of linear combinations of undetermined functions and their derivatives up to a certain order, referred to as their rank. The article presents an algorithmic method for determining finite-rank solutions for linear PDEs of arbitrary order and with any number of independent variables—representing a significant generalization of Laplace's method. This approach is developed in detail for Euler-Poisson-Darboux equations with one, two, or three spatial variables. Several solutions are explicitly provided and compared with so-called complete solutions. Furthermore, the extension of this method to general linear PDEs is discussed.

Abstract: This paper investigates Devaney chaos in the \(L^{1}\)-function envelope system associated with a nonautonomous discrete dynamical system. We prove that if (I, f_∞) is weakly mixing of order 3, then its \(L^1\)-functional envelope system is weakly mixing of all orders. Moreover, if, in addition, the periodic points of (I, f_∞) are dense in \(I\), then the functional envelope system \((L^1(I,I),H_{\infty})\) is Devaney chaotic.

Abstract: Banach spaces play a fundamental role in functional analysis and provide the standard analytical framework for a broad range of physical theories. Their importance, however, raises a natural mathematical question: does the existence of a continuous physical evolution logically entail Banach completeness of the underlying normed realization? In this paper, we address this question within a general functional-analytic framework. We first prove that every dense invariant normed realization possesses a unique invariant closed extension. We then establish that every continuous family of evolution operators admits a unique extension to the Banach completion of the underlying normed realization. These structural results yield the principal theorem of the paper: continuous evolution does not entail Banach completeness. Banach completion preserves the operator family uniquely, but completeness itself is not a logical consequence of the dynamics. The general theorem is subsequently verified for representative mathematical formulations of classical Newtonian dynamics, Schr\"odinger quantum dynamics, the Cauchy formulation of the Einstein equations, the classical gauge-fixed Polyakov formulation of perturbative string theory, and the classical low-energy eleven-dimensional supergravity formulation associated with M-theory. The results identify Banach completion as a canonical analytical closure procedure rather than a property forced by the underlying evolution laws---thereby separating the analytical utility of completeness from its logical status in the mathematical formulation of physical theories.