Computer Science and Mathematics

Abstract: In this paper, we obtain new criteria for the norm and the essential norm of composition operators acting from the space \(\mathcal{H}^{\infty}\) of bounded harmonic mappings into weighted harmonic Bloch spaces \(\mathcal{B}^{\alpha}_H\), \(0< \alpha< \infty\), on the unit disk. These criteria are expressed in terms of the asymptotic behavior of the sequence \(\{\|C_{\psi}q_k\|_{\mathcal{B}^{\alpha}_H}\}_{k\ge0}\) and of a family of M\"obius-invariant test functions, yielding symbol-based characterizations of boundedness and of compactness, and hence of the essential norm for \(C_{\psi}:\mathcal{H}^{\infty} \to \mathcal{B}^{\alpha}_H\).

Abstract: For a real correlation matrix \(C\), the self-conjugate specialization of Chollet's conjecture asserts \(\operatorname{per}(C\circ C)\leq \operatorname{per}(C)^2\). We determine the sharp constant in order five under the rank-two constraint: \begin{array}{c} \operatorname{per}(C\circ C)\leq \dfrac{137}{1440}\,\operatorname{per}(C)^2,\\ C\in\mathbb{R}^{5\times5},\qquad C\succeq0,\qquad \operatorname{diag} C=\mathbf1,\qquad \operatorname{rank} C\leq2. \end{array} Equality holds exactly when the five Gram lines form a regular pentagon in \(\mathbb{RP}^1\), up to permutation and sign switching. The proof reduces the two permanents to elementary symmetric functions of five unimodular numbers, solves the remaining two-phase minimization, and ends with explicit sum-of-squares and positive-semidefinite Gram certificates. We also prove a uniform order-five result without a rank assumption: Chollet's inequality is strict away from the identity whenever \(\lambda_{\min}(C)\geq3/4\). All finite algebraic certificates are supplied in exact arithmetic.

Abstract: In the article, a subclass of Starlike functions linked with Bernoulli,s lemniscate is studied in the unit disc. The bounds of H_3,2 Hankel determinant, the sixth coefficient |b₆| and the Zalcman functionals for the class SL* are obtained. The results are sharp.

Abstract: For a graph G, let \( \nu_{\triangle}(G) \) be the largest number of pairwise edge-disjoint triangles and let \( \tau_{\triangle}(G) \) be the smallest number of edges meeting every triangle. Tuza's conjectured that \( \tau_{\triangle}(G)\leq 2\nu_{\triangle}(G) \) for every graph. We prove the conjecture for every split graph with a specified split partition \( V(G)=C\mathbin{\dot\cup}I \) such that \( |C|=8 \), for which the triangle-active vertices of I have at most two distinct neighborhoods in C. The result allows arbitrary multiplicities of the two types. The infinite statement is reduced to 44,702 canonical cases by an exact multiplicity-truncation lemma and symmetry. Each case carries an explicit triangle cover and an explicit edge-disjoint triangle packing, checked by an independent standard-library verifier. We also give exact cover and packing reductions for arbitrary split graphs and a maximum-cut criterion that settles, at \( |C|=8 \), every instance whose independent-side neighborhood 2-shadow has at most seven edges. The unrestricted split-graph and interval-graph cases remain open.

Abstract: For the intersection of two disks meeting at angle 2α, let C(α) be the least constant in the associated spectral-set inequality, uniformly over operators for which each disk separately is a spectral set. The exact value of C(α) is unknown except at the disk endpoint α = π/2. We give a self-contained Möbius reduction to the corresponding numerical-range problem on a sector and compute the sharp constant on the infinite-dimensional class of affine square-zero operators B = λI + N, N² = 0: C_sq0(α) = πsin α/2α. A 2 × 2 matrix and a conformal extremal attain equality and yield an explicit lens lower-bound certificate. At the right angle we prove the conjectural 2 estimate, in arbitrary dimension, for the full palindromic quadratic family. Exact rational matrix certificates further cover the complex post-automorphism disk |c| ≤ 19/20, the complete imaginary diameter and a transverse cusp, and boundary-reaching phase arcs whose union misses only 9.17 degrees of the parameter circle near -1. For every symmetric three-node set in the right-angle disk coordinate, we also prove the sharp identity-multiplier estimate on the complete admissible-kernel cone; the proof combines an exact extreme-ray rank bound, scalar Pick interpolation on the rank-one faces, and a two-variable Bernstein certificate on the rank-(2, 2) face. A nested exact certificate extends this result to the genuinely asymmetric two-parameter patch ρ = (−a, 0, b), 1/6 ≤ a, b ≤ 5/6. On a rank-one localized face we also prove a three-real-parameter singular-boundary family: for Φ(c) = e^iθc², sin θ > 1/√2, the sharp target is positive for every quadratic Blaschke product w(w − a)/(1 − a¯w) with a ∈ D. At the central square, a polynomial bidisk extension and Ando’s theorem prove ∥w²∥ ≤ κ₀ < √2, where 276889/127200 − 6103√2/10600 = 1.362560 . . .. We also derive an exact two-complex-parameter consequence: the same construction proves a strict √2 bound for every b_αb_β with |α|, |β| ≤ 1/150, allowing two independently phased nonzero zeros. We also derive an exact two-moment criterion for the remaining boundary layer and a verified angle-dependent envelope. Every computer-assisted assertion has an exact rational verifier. These results are dimension-free but do not determine the unrestricted fixed-lens constant.

Abstract: For a real correlation matrix C, the self-conjugate specializa- tion of Chollet’s conjecture asserts per(C ◦ C) ≤ per(C)2. We determine the sharp constant in order five under the rank-two constraint: per(C ◦ C) ≤ 137 1440 per(C)², C ∈ R^5×5, C ⪰ 0, diag C = 1, rank C ≤ 2. Equality holds exactly when the five Gram lines form a regular pentagon in RP¹, up to permutation and sign switching. The proof reduces the two permanents to elementary symmetric functions of five unimodular numbers, solves the remaining two-phase minimization, and ends with explicit sum-of-squares and positive-semidefinite Gram certificates. We then study the rank-three stratum. The matrix J2 ⊕ ( 1 −1/2 −1/2 −1/2 1 −1/2 −1/2 −1/2 1 ) has self-Chollet ratio 13/48. We prove that this value is sharp in two infinite structured rank-three families and that the displayed configura- tion is a strict local maximum in the full rank-three configuration space. We also prove Chollet’s inequality whenever |cij | ≤ 27/50 for i̸ = j, with the sharper estimate R(C) ≤ 621/800 at coherence 1/2. Finally, without a rank assumption, Chollet’s inequality is strict away from the identity whenever λmin(C) ≥ 3/4. The proposed global rank-three value 13/48 remains open. All finite algebraic certificates are supplied in exact arithmetic.

Abstract: Let \( p\geq 5 \) be prime. The theorem of Yu obtained from permanent rank proves the Alon--Jaeger--Tarsi conjecture over \( F_p \) in dimensions \( n<2^{p-2} \). We close the missing boundary: the conjecture holds for every \( n\leq 2^{p-2} \). In particular, for every \( M\in GL_8(F_5) \) there is a vector \( x\in(F_5^*)^8 \) such that \( Mx\in(F_5^*)^8 \). The proof separates according to the permanent rank of \( M \). Above half rank, Yu's concatenation recurrence and the Combinatorial Nullstellensatz force a nowhere-zero point. At equality, the rigidity theorem of Kisley and Shader reduces \( M \), under monomial equivalence, to a direct sum of \( 2\times2 \) blocks, for which a nowhere-zero point is explicit.

Abstract: We solve, for the complex spaces \( \ell_\infty^2 \) and \( \ell_1^2 \), the existence problem for normalized equiangular tight systems posed by K. Mahesh Krishna. We first prove a rigidity statement independent of tightness: if all off-diagonal values have a common modulus \( 0\leq\gamma<1 \), then a normalized system in \( \ell_\infty^2 \) has at most three members. In the tight range \( 0<\gamma<1 \), existence is therefore equivalent to\( n=3 \) and \( \gamma=1/2 \), and we classify every extremal system. We then classify the endpoint \( \gamma=1 \): such a tight system exists exactly when \( n\geq4 \) is even. Since \( \gamma>1 \) is impossible, this gives the complete list of admissible pairs \( (n,\gamma) \) for both spaces. Explicit parametrizations and constructions are provided, and the \( \ell_1^2 \) results follow through a duality principle that preserves normalization, equiangularity, and tightness.

Abstract: For m ≥ 2, let cp(m) be the all-dimensional best constant in ||m∑k=1 Ak||p ≤ cp(m) || m ∑k=1|Ak| ||p. Tang and Zhang conjectured an explicit formula for every finite p > 1. We disprove the conjecture with two explicit real 2 × 2 rank-one matrices at p = 3/2. The comparison is certified by seven strict rational inequalities and, in particular, places the attained ratio above 207/200 while the conjectured constant lies below 207/200. On the positive side, we prove the conjectured sharp bound for every family of rank-at-most-one summands when 2 ≤ p < ∞, and classify all equality cases. We also prove the corresponding endpoint statement for p = ∞. Finally, for arbitrary complex matrices we establish the conjectured sharp constant in the case m = 2, p = 4.

Abstract: Employing the fundamental techniques of substitution (change of variables) and integration by parts, the discussion in this letter focuses on obtaining explicit formulas of the integral\( K_{p,r} =\int x^p \left ( 1+ x^2\right )^r dx \) mainly for \( p \) an integer and \( r \) an integral multiple of \( \frac{1}{2} \). As the case where \( r \) is a non-negative integer is rather simple and obvious, we concentrate mainly on the cases where \( r \) is a negative integer or an odd multiple of \( \frac{1}{2} \) except when consideration of other cases is appropriate.

Abstract: We develop a comprehensive Lagrangian framework for the analysis of singularities in the three-dimensional compressible rotating chemotaxis–Navier–Stokes system, with particular emphasis on the high Mach number regime. Focusing on suitable weak solutions that satisfy the entropy inequality, we introduce the notion of Lagrangian singular trajectories adapted to the compressible setting and establish a geometric characterization of the space–time blow-up set. Our main theoretical advance shows that singularities are confined to a low-dimensional Lagrangian structure transported by the flow, even in the presence of strong acoustic waves and rotational effects. More precisely, we prove that the space–time singular set is contained in a countable union of Lagrangian trajectories associated with the velocity field and satisfies the sharp estimate that its Hausdorff dimension is at most one. This result constitutes a substantial refinement of classical Eulerian partial regularity bounds of Caffarelli–Kohn–Nirenberg type and provides a genuinely geometric interpretation of singularity formation in coupled fluid–chemotaxis models under extreme compressibility and rotation. The proof combines global entropy inequalities, compactness methods, and partial regularity theory with a refined analysis of the Lagrangian flow map in the DiPerna–Lions–Ambrosio setting for transport equations with variable density. A key feature of our approach is the propagation of regularity along particle trajectories weighted by the density, which allows singularities to be tracked dynamically and yields improved dimensional estimates via tools from geometric measure theory. Additionally, we establish a Lagrangian regularity criterion expressed solely in terms of the integrability of the velocity along particle trajectories, providing a sufficient condition for global smoothness. Beyond the dimensional bound, the proposed Lagrangian formulation clarifies the mechanism by which chemotactic forcing interacts with compressible fluid transport and rotation to produce potential blow-up and establishes a direct connection between singularity formation and low-dimensional invariant structures. These results open new perspectives for the geometric analysis of singularities in active fluid systems and related nonlinear partial differential equations.

Abstract: The usual power series \(\sum\alpha_{n}x^{n}\) are generalized by the substitution $x\to g(x)$ to power series built from a function \(\sum a_{n}g^{n}(x)\). The key feature of our approach consists in using functions \(g\) with free parameters \(g(x)=g(\{p_{i}\},x)\); the resulting expansions thus keep the Taylor‐like behavior (i.e., they match derivatives at the expansion point), but can also be tuned to adjust the far‐away behavior. We present six specific cases \(g_{X}\), $X\in\{A,\dots,F\}$, each containing five free parameters. We use examples to demonstrate that the generalization can improve the convergence (rate and domain), mimic branch points and branch cuts and have other interesting properties. Because the partial Bell polynomials are a necessary ingredient in our method, our results can be interpreted as new formulas for specific values of the Bell polynomials.

Abstract: This paper investigates the existence of multi-bubble solutions for a class of critical elliptic systems in the physically relevant dimension N = 3. While the Lyapunov-Schmidt reduction method has been successfully applied to scalar equations with competing potentials, its extension to coupled systems remains largely unexplored. The present work fills this gap by developing a rigorous reduction framework for a coupled system featuring a subcritical coupling term of order 2 < p < 6, which is treated as a lower-order perturbation of the critical elliptic structure. The construction proceeds in several stages. First, we introduce suitable weighted norms tailored to capture the multi-scale interactions between bubbles arranged on a symmetric circular lattice. Second, we decompose the linearised operator into a diagonal part, corresponding to the scalar critical operator, and a coupling part, which is shown to be a contraction for large dilation parameters due to the subcritical nature of the coupling. Third, we solve an auxiliary projected problem via a contraction mapping argument, obtaining a unique remainder term controlled in the weighted norms. The key novelty lies in the use of local Pohozaev identities to eliminate the Lagrange multipliers that enforce orthogonality, thereby reducing the problem to finding critical points of a finite-dimensional energy functional. Our main result establishes that, under natural symmetry and non-degeneracy assumptions on the potentials, for any sufficiently large integer m there exists a positive solution consisting of m interacting bubbles. The dilation parameters satisfy the precise scaling law λ_m = Cm + o(m), where the constant C is determined by a balance between self-energy and repulsive interaction. The energy of these solutions grows linearly with m, namely I(u_m, v_m) = m(A₀ + o(1)), where A₀ > 0 is the energy of a single bubble in the homogeneous case. The remainder terms are small in the weighted norms, ensuring the validity of the asymptotic expansion. These results contribute to the understanding of concentration phenomena in coupled elliptic systems and have potential applications in nonlinear optics and Bose-Einstein condensates, where similar systems arise naturally.

Abstract: In this paper, we study coefficient problems in some subclasses of convex functions. More precisely, we determine the upper bounds of initial coefficients \( |a_i|(2\leq i\leq 6) \), Zalcman inequalities, the second and third Hankel determinants, the second-order Hankel determinant of logarithmic coefficients and the third Hankel determinant of inverse functions for the class \( \mathcal CL \). All of the bounds are sharp.

Abstract: We investigate a time-fractional telegraph equation with variable coefficients involving Hilfer fractional derivatives. The problem is formulated as a Cauchy problem with Hilfer-type initial conditions. By applying the Fourier transform with respect to the spatial variable, we reduce the equation to a Hilfer-type fractional differential equation with variable coefficients. We then derive an explicit representation formula expressed in terms of convergent Neumann-type series involving compositions of generally non-commuting operators. Particular attention is devoted to the power-law and constant coefficients cases, for which more explicit formulas are obtained. By inverting the Fourier transform, we derive distributional space-time representations of the solution in terms of space-time convolutions and Neumann-type operator series. In the constant-coefficient case, the obtained formulas recover several known representations available in the literature.

Abstract: The integration of artificial intelligence (AI) into health systems requires instruments capable of diagnosing student performance beyond global scores, explaining why a failure occurs and how to remediate it, without delegating decisions to opaque algorithms. This study, guided by Design Science Research, presents the design and specification of the LIPS platform (Integrated Laboratory of Simulated Practices), a closed-loop, seven-layer sociotechnical architecture for the granular assessment of competencies in health professions education. The architecture decomposes clinical activities into observable micro-skills, paired with assessment items through a Q-matrix, and isolates the deterministic computation of scores from interpretation by generative AI, restricted to the formulation of explanatory hypotheses linked to evidence and submitted to instructor validation (human-in-the-loop). The demonstration applied the method to an illustrative airway suctioning scenario: the decomposition generated ten micro-skills and twenty items fully linked through the Q-matrix, and the diagnostic pathway showed how a global performance of 90% would mask a critical safety failure localized in a specific micro-skill. The formative evaluation is presented as a structured protocol, prior to application with human participants. The central contribution lies in the method for granularity of competencies and for containment of generative AI, transferable to other health professions education systems.

Abstract: Let \[ F(z):= (z+2)\zeta(z+1)\zeta(z+3) -(z+1)\zeta^2(z+2) -\zeta(z+1)\zeta(z+2), \qquad z>0. \] The positivity of \(F\) for positive real parameters arises as an extension of an inequality previously established for positive integer arguments. In this paper, we first introduce the tail function \[ \tau(s)=\zeta(s)-1 \] and derive the general lower bound \[ F(z)\ge \bigl(z-\tau(z+1)\bigr) \bigl(\tau(z+2)-\tau(z+3)\bigr), \qquad z>0. \] Consequently, \(F(z)>0\) whenever \[ \zeta(z+1)0,\qquad z>z_0. \] Numerically, \[ z_0\approx0.8337726517. \] We next introduce \[ H(x)= x\left( 1-\frac{\zeta(x+1)}{\zeta(x)} \right), \qquad x>1, \] and establish the exact identity \[ F(z) = \zeta(z+1)\zeta(z+2) \bigl(H(z+1)-H(z+2)\bigr). \] Thus the original positivity problem is equivalent to the unit-step inequality \[ H(x)>H(x+1), \qquad x>1. \] We further investigate known monotonicity results for shifted ratios of Riemann zeta values. The weighted shifted-ratio theorem of Guo and Qi provides useful monotonicity information but does not by itself imply the required unit-step inequality. A stronger two-variable zeta-ratio monotonicity theorem of Yang and Tian, when applied with the precise normalization considered below, yields the estimate \[ \frac{\zeta(x+2)}{\zeta(x+1)} > \frac{2}{ 3-\zeta(x+1)/\zeta(x) }, \qquad x>1. \] An elementary comparison shows that this estimate is sufficiently strong to imply the required unit-step inequality for \(H\) and, consequently, the positivity of \(F(z)\) for every \(z>0\).

Abstract: We introduce spectral Weyl spaces, a new class of Banach spaces that incorporate the discrete spectral structure of the Schrödinger operator associated with the Korteweg–de Vries (KdV) equation. These spaces naturally separate the solitonic and radiative components of solutions, providing a setting where the influence of bound states on regularity is explicitly quantified. Our main theorem establishes global well-posedness of the KdV equation in these spaces for regularity indices s > 1/2, with a sharp polynomial growth estimate for the Hs-norm. The growth constant depends explicitly on a spectral functional that sums over the negative eigenvalues of the Lax operator, precisely capturing the contribution of solitons to the growth of higher-order Sobolev norms. As corollaries, we obtain the invariance of the number of solitons, an asymptotic decomposition into solitonic and radiative parts, and improved bounds for purely radiative data. In the second part, we construct a formal quantum neural network operator (QNNO) that approximates the KdV flow in finite-dimensional truncations. We prove a complete asymptotic expansion for the approximation error, revealing a three-layer structure: integer powers from ordinary Fréchet derivatives, fractional powers governed by Marchaud derivatives that capture Hölder smoothness, and purely quantum commutator terms arising from the non-commutativity of the Lax pair. The remainder is bounded by an explicit constant. This work establishes a rigorous bridge between classical dispersive PDE theory, spectral analysis, fractional calculus, and quantum machine learning.

Abstract: The Fermat theorem that is often referred to as the interior extrema theorem is one of the fundamental results in calculus and optimization theory. Recently, the statement of this theorem has been generalized to the case of fractional derivatives. Unlike in the classical case, these fractional analogs typically appear in the form of inequalities rather than equalities. Moreover, the exact form of these inequalities depends on the particular definition of the fractional derivative being used. In this paper, for the first time, we present Fermat-type results for the 1st level general fractional derivative that has the Caputo, Riemann–Liouville, and Hilfer derivatives, as well as the general fractional derivatives and the regularized general fractional derivatives with Sonin kernels among its particular cases. We also discuss some applications of this fractional analogy of Fermat’s theorem including derivation of a comparison principle for the fractional differential inequalities involving the 1st level general fractional derivatives as well as a priori estimates for solutions of the initial-value problems for the fractional differential equations with the 1st level general fractional derivatives.

Abstract: This paper develops a theory of fractional Landau inequalities in mixed Sobolev norms, extending classical derivative estimates to anisotropic function spaces. We introduce mixed fractional Sobolev spaces \( W_{\alpha}^{\nu,p}(\mathbb{R}^{k}) \), where \( \alpha = (\alpha_1,\dots,\alpha_k) \) encodes directional scaling and characterizes functions with coordinate-dependent regularity. Within this framework, we establish fractional Landau inequalities with constants depending explicitly on the fractional order ν and the anisotropy vector α. The analysis relies on techniques from anisotropic harmonic analysis, including directional Littlewood–Paley decompositions and anisotropic maximal function estimates. The sharpness of the inequalities is discussed, and connections to approximation theory are outlined. These results provide a mathematical bridge between fractional calculus, harmonic analysis, and high-dimensional approximation.