Papers

The preprint arXiv:2609.04176 (Z.-W. Sun, 3 September 2026) claims a proof that Catalan's constant $G$ is irrational: it constructs for each $B$ a number $\hat q_B$ from weighted tails of the series and asserts that, if $G=a/q$, the integer $N_B=q^S H_B^{\min}\hat q_B$ satisfies $0<|N_B|<1$ for large $B$ (its Theorem 9.1). We report four exact computations. First, the printed proof of the rank theorem (Theorem 2.1) carries $T_i$ where its own recurrence requires $T_{i+1}$; the slip is repairable, and the statement is verified exactly at 44 parameter pairs with $S\le 4$, $B\le 30$. Second, with the paper's definitions and the worst-case denominator, attained at an explicit rational $a/q$, a lower bound for the quantity the derivation of Theorem 9.1 controls equals $+1.49$ to $+1.79\,B^2$ at eighteen indices $20\le B\le 119$, where the derivation claims at most $-0.00966\,B^2+o(B^2)$. Third, the paper's own denominator bound (Corollary 5.2) holds in every tested instance and is nearly sharp, so the discrepancy lies in the asymptotic ledger of Sections 6-9. Fourth, an exact identity locates it: the prime 2 contributes $v_2(F_D)\log 2=(2\log 2+o(1))B^2$ to the quantity, because the positive part at 2 vanishes while $|\hat q_B|$ carries the full power of 2 in $F_D$; no display in the ledger carries a term of this order, and Remark 9.3 assigns it to the derivation of the odd-prime constant $c_{\rm odd}=0.006$ without showing how. At the tested parameters the odd-prime layers agree with the paper's constants to within about $0.1\,B^2$, and the remainder of the discrepancy is the size of the residual minor at those indices. These computations test the paper's bounding method, which is uniform in $(a,q)$; they show that the published derivation does not justify the claimed estimate, so the proof of the main theorem is incomplete. Nothing is concluded about the irrationality of $G$.

We prove the irreducibility conjecture for Legendre polynomials proposed by Stieltjes: for every integer $j\geq 1$, both $P_{2j}(x)$ and $P_{2j+1}(x)/x$ are irreducible over the rational numbers. The proof proceeds by contradiction. Starting from a hypothetical nontrivial factorization, we remove the zero root from $P_n$ when present, multiply by a suitable constant, and express the resulting polynomial as a product of two integral polynomials $A(x^2)$ and $B(x^2)$. We then construct the resultant and its odd part, $R=\operatorname{Res}_t(A,B), \mathcal{R}=\frac{|R|}{2^{v_2(R)}}.$ Put $k=\operatorname{deg} A\leq\operatorname{deg} B$ and $m=\lfloor n/2\rfloor$. Using the classical auxiliary polynomial $U_n$ and the differential identity $(1-x^2)(P_n'U_n-P_nU_n')=1-P_n^2,$ together with orthogonality, divisibility properties of the coefficients, and a least-common-multiple estimate, we obtain an upper bound for $\mathcal{R}$. On the other hand, the Legendre differential equation gives a lower bound for the absolute value of the derivative of $AB$ at each root of $A$. We use Chebyshev polynomials and Hadamard's inequality to bound the product of the squared pairwise differences of these roots, and combine this with estimates for the leading coefficients and the power of $2$ in $R$ to obtain a lower bound for $\mathcal{R}$. These estimates yield $m\log4-\log\frac{4(m+1)^2}{\sqrt m} <\frac{\log\mathcal R}{k} <(m+\sqrt{2n-1})\log3 (n\ge64).$ For every $n\ge256$, the lower bound strictly exceeds the upper bound, giving a contradiction. Combining this with established irreducibility results and explicit integer comparisons for the remaining degrees proves irreducibility in every degree.

We prove Stieltjes' common root conjecture: Legendre polynomials of distinct degrees have no common nonzero root. We construct an auxiliary polynomial and show that, for the relevant power of two $q\ge2$, the edge of slope $1/q$ in its $2$-adic Newton polygon has horizontal length less than $3q$. If a common nonzero root existed, Newton polygon theory and the Legendre differential equation would force the same edge to have horizontal length at least $3q$, yielding a contradiction.

We conjecture an explicit evaluation of the pure fifth Bessel moment $\int_0^\infty K_0(t)^5\,dt$ as a quadratic expression in the critical value $L(f,2)$ of the weight-three, level-60 newform $f$ (LMFDB orbit 60.3.b.a) identified in the twisted fifth-moment modularity theorem of Lim, Tu and Yu, with coefficients in $\mathbb{Q}(\sqrt{5})$ and the square taken before the real and imaginary parts. Directed interval computations, using no stored Bessel or $L$-values, bound the absolute discrepancy by $10^{-358}$. We prove three exact modular identities for $f$: the Petersson-norm formula $\langle f,f\rangle_{60} = \frac{3(5-\sqrt{5})}{2\pi^4}|L(f,2)|^2$, the coefficient-conjugation relation $L(f^{\sigma},2) = \kappa L(f,2)$ with explicit $\kappa \in \mathbb{Q}(\sqrt{5},i)$, and the twisted symmetric-square evaluation $L(\chi_{-4}\mathrm{Sym}^2 f,2) = \sqrt{15}\,\pi^2 \langle f,f\rangle_{60}$, together with $L(\chi_{-4}\mathrm{Sym}^2 f,3) = \pi^4\langle f,f\rangle_{60}/8$, in the full Euler-factor normalization of Lim, Tu and Yu. The last identity shows that the companion norm conjecture $D_{5,\mathrm{odd}} = \frac{3\sqrt{15}(5-\sqrt{5})}{2}|L(f,2)|^2$ is equivalent to the symmetric-square conjecture $D_{5,\mathrm{odd}} = \pi^2 L(\chi_{-4}\mathrm{Sym}^2 f,2)$ of Lim, Tu and Yu, while the exact relation $D_{5,\mathrm{even}} = \pi^2 D_{5,\mathrm{odd}}/(2\sqrt{15})$ follows from Chuang's period formulas. Every Bessel-to-modular equality, including the individual-period formula, remains conjectural. Complete proofs, exact rational certificates and verification programs are included as ancillary files.

In this article, we prove the irrationality of at least one of the three numbers $\zeta(5), \zeta(7)$ and $\zeta(9)$ is irrational. We find linear forms in $1,\zeta(5),\zeta(7)$ and $\zeta(9)$ and prove their arithmetic by Kranttenthaler and Rivoal. We also refine these estimates by using integral quantity $\Phi$ involving specific product of prime powers. We use the results of Kranttenthaler, Rivoal and Zudilin to prove that at least one of the three numbers $\zeta(5), \zeta(7)$ and $\zeta(9)$ is irrational.

In this paper, the unified generalized $q$-Stirling matrices are defined in terms of the $q$-analogues of the unified generalized Stirling numbers. Several matrix relations involving these matrices are derived, and matrix decompositions of the unified generalized $q$-Stirling matrices are established.

Let $5\leq p_1<\cdots<p_r$ be primes, put $P=p_1\cdots p_r$, and let $g_P(2n)$ count the unordered decompositions $2n=h+k$ for which both summands are coprime to $6P$. We derive an explicit finite inclusion–exclusion formula for $g_P(2n)$ by parametrizing the admissible residue classes and combining the local restrictions through the Chinese remainder theorem. The associated complete-modulus correlation is a specialization of Nagell's totient function and determines both the finite local density and the affine increment $g_P(2(n+6P))-g_P(2n)=\kappa_P(n)$. A parameter-density decomposition gives a residue-sensitive positivity criterion and the uniform sufficient condition $n\geq 3(3^r-1)\prod_{i=1}^{r}\frac{p_i}{p_i-2}+2\Longrightarrow g_P(2n)>0$. Maximizing this bound over squarefree moduli whose prime factors do not exceed a prime $p$ gives an explicit threshold $\mathcal N(p)$ satisfying $\log\mathcal N(p)\sim(\log3)p/\log p$. We also describe the complete prime cutoff at $\sqrt{2n}$ and its exact additive-prime interpretation. We also evaluate $g_P(2tP)$ explicitly using the real character of period $3$. The positivity thresholds are elementary and fully explicit; lower-bound sieves provide smaller asymptotic scales, and the present thresholds do not establish positivity at the complete cutoff for all $n\geq13$.

Let ${\mathbb K}$ be an arbitrary field. We study tensor degenerations for square matrix multiplication over the field ${\mathbb K}$, and mainly prove for any integer $n\ge5$ the uniform dyadic estimate \[ \underline{\mathrm R}_{\mathbb K}({\mathsf M}_{2^n}) \le 2^{2n+1}\] which implies that $\omega(\mathbb K)\le 2+1/n$, where the border rank $\underline{\mathrm R}_{\mathbb K}(T)$ is the least $r$ for which the scalar extension of $T$ to an algebraic closure of $\mathbb K$ lies in the Zariski closure of the tensors of rank at most $r$, and $\omega(\mathbb K)$ is the matrix multiplication exponent over the field ${\mathbb K}$. This improves the current best record $\omega(\mathbb K)<2.371177$ greatly, and answers affirmatively the long-standing open question whether the value $\omega({\mathbb K})$ is exactly $2$. The implementation disclosure and appendices in this paper give fixed source maps, interpolation data, terminal incidence ledgers, and deterministic check counts for independent reconstruction.

In this article, we prove two conjectures of Sun concerning the evaluation of two series involving the reciprocal of the binomial coefficient $\binom{4k}{k}$. The values of these series are related to $\pi$, $\log3$, and the Gieseking constant $\text{Cl}_2(\pi/3)$. Our proofs rely on the Beta function and the Clausen function.

For an odd integer $q\geq3$, let $a_q(r)$ count finite words in $\{q/2,1/2\}$ containing exactly $r$ factors $q/2$, with every proper prefix product greater than $1$ and total product less than $1$. For $q=3,5,7$, these are OEIS A100982, A174795, and A174796. The increasing-sequence representation yields a range identity and, for A100982, an exact recurrence valid at every order. Using the classical cycle lemma, we prove the natural lower and upper binomial bounds and characterize equality. The equality orders are precisely the strict lower and upper one-sided best-approximation denominators to $\log_2q$, equivalently the convergent and semiconvergent denominators and the record approaches of $q^r$ to powers of $2$. At every nontrivial equality order, the admissible words are in bijection with rational Dyck paths and their number is a rational Catalan number. We also prove monotonicity in $q$, determine the exponential growth constants, and describe the exact normalized oscillation.

Let $h(n) = n + \tau(n)$, where $\tau$ counts divisors. Erdős and Graham asked, after Spiro, whether the orbits of any two positive integers under $h$ eventually share a point (Problem 414 on Bloom's list). The problem is open, and a finite computation cannot close it; nothing about it has been checked by a proof kernel, and the statement in the formal-conjectures repository carries a sorry. We replay what can be certified through the Lean 4 kernel under an axiom gate (axioms exactly propext, Classical.choice, Quot.sound; no sorry; no native_decide). Three elementary facts recorded by Li, namely that coalescence is an equivalence relation, $\tau(n)$ is odd exactly for squares, and $\tau(n)$ is bounded by $2\sqrt{n}$ so that orbits skip no square annulus $[k^2, (k+1)^2)$, are formalized, and his frontier lemma is formalized in a window form: every orbit started below a level $N$ has a point within $\lceil 2\sqrt{N} \rceil$ below $N$. The frontier lemma becomes a certificate: the $\lceil 2\sqrt{N} \rceil$ values $\tau(m)$ just below $N$ and the orbit points above $N$ until the orbits through the crossing points have merged. Its soundness is a kernel-checked theorem, and the certified rungs state that every pair of positive integers below $N$ coalesces for $N = 10^5, 10^6, 10^7, 10^8$; the certificate at $10^8$ has 44,530 entries rather than $10^8$. Every ledger and every quoted count was produced by two programs sharing no code that agree by hash. We record what is known about a non-coalescing pair and measure the exit-set sizes Li bounds. Nothing here is a proof of the conjecture.

This paper introduces a novel class of non-linear multiorder fractional differential operators. First, we provide intuitive and formal justifications for the existence of multiorder differential operators through differential analysis and Omega-derivative formulations. We then define the conformable multiorder derivative as a functional quotient of fractional derivatives and establish its analytical properties rigorously. Furthermore, leveraging fundamental principles of mathematical analysis, we construct its inverse operator, termed the conformable multiorder integral. Detailed step-by-step proofs of all theoretical properties are presented. Illustrative examples and applications to solving elementary multiorder differential equations are thoroughly examined. Finally, comprehensive conclusions and strategic outlooks for prospective research directions are discussed.

Let \(g_n(x)\) be the Mittag–Leffler polynomials defined by $$ \sum_{n\ge0}g_n(x)t^n=\frac12\left(\frac{1+t}{1-t}\right)^x. $$ We derive the coefficient formula $$ g_n(x)=\frac1n\sum_{j=0}^{\lfloor (n-1)/2\rfloor} (-1)^j\frac{2^{n-2j-1}}{(n-2j-1)!} [u^{2j}](u\cot u)^n\,x^{n-2j}, $$ equivalently $$ [u^{n-r}](u\cot u)^n =(-1)^{(n-r)/2}\frac{n(r-1)!}{2^{r-1}}[x^r]g_n(x). $$ The identity is suggested by comparing an umbral representation of the hyperbolic tangent integral $$ K(a,b)=\int_0^\infty \frac{\tanh^a x}{x^b}\,dx $$ with a compact cotangent-coefficient formula, but is proved independently from the generating function by formal Lagrange–Bürmann inversion. Substitution of the proved bridge back into the integral formula then yields a rigorous derivation of the umbral Mittag–Leffler representation. We also develop a Dirichlet–beta analogue for $$ I_{m,n}^{\beta}=\int_0^\infty\frac{\tanh^m x}{x^n\cosh x}\,dx. $$ Defining \(q_m(x)\) by $$ \sum_{m\ge0}q_m(x)t^m=(1-t^2)^{-1/2}\exp(2x\operatorname{artanh}t), $$ we prove $$ [u^{m-r}]\frac{u}{\sin u}(u\cot u)^m =(-1)^{(m-r)/2}\frac{r!}{2^r}[x^r]q_m(x). $$ This gives a parallel umbral representation for the beta integrals. The family \(q_m\) is a shifted \(c=-1,\beta=1\) specialization of the classical Meixner family, yielding a beta/Meixner counterpart to the zeta/Mittag–Leffler correspondence.

The decimal reversals 1729 and 9271 have multiplicative orders of $2$ equal to 36 and 63. Motivated by this symmetry, we classify the affine commutation equation $R_B(hx+1)=hR_B(x)+1$, where $R_B$ reverses base-$B$ digits, for every $B\ge3$, $2\le h<B$ and positive integer $x$ such that neither $x$ nor $hx+1$ ends in zero. Comparing the two carry sequences separates the solutions by their digit lengths. Equal lengths force an odd number of digits alternating between two explicit intervals. When the output gains a digit, coprimality of $h$ and $B+1$ forces a repdigit input with palindromic output. In the noncoprime case, all inputs instead have even length and form a finite family. A slack-variable bijection gives the sharp maximum length, binomial counts at each length, and totals expressed through Fibonacci numbers or powers of two. For reversal pairs of primes already known to share an order index $2\le h<B$, the classification gives an exact digit criterion for reversal of their orders. Finally, an exhaustive computation with independently checkable primality certificates proves that $\{1729,9271\}$ is the unique pair of distinct odd decimal reversals whose orders of $2$ are distinct two-digit reversals of each other.

In a previous article doi.org/10.3390/geometry2030012, the author investigated a class of elementary plane geometry problems closely related to the theme of this work. Here, we prove a weaker version of a previous conjecture by demonstrating that there are infinitely many maximally circlable (MAC) numbers – positive integers $n$ for which there exists a largest circle enclosing exactly $n$ interior lattice points. Furthermore, by extending numerical computations to $n \le 2700$, we identify two counterexamples to a conjecture in loc cit. regarding the symmetry of the largest circle enclosing a strong MAC number (a MAC number $n$ where $n+1$ is non-MAC). We also propose a potential infinite family of strong MAC numbers derived from Pythagorean triples; the existence of this family would imply the infinity of non-MAC numbers, as conjectured by Zhao. Throughout this paper, we provide extensive data characterizing both MAC and strong MAC numbers alongside their corresponding largest enclosing circles.

This paper investigates the completeness properties of the fuzzy order on the fuzzy number space $\mathbb{E}^1$ within the framework of $[0,1]$-enriched categories. \(S\)-convergence under the three fundamental continuous t-norms is characterized, and inclusion relations among the induced open ball topologies are established. It is shown that $(\mathbb{E}^1,P)$ is Cauchy complete if and only if the t-norm is not isomorphic to the Łukasiewicz t-norm, whereas $(\mathbb{E}^1,P)$ fails to be Smyth complete for every continuous t-norm. Consequently, the symmetrization $(\mathbb{E}^1,S)$ is Smyth complete if and only if the underlying continuous t-norm is not isomorphic to the Łukasiewicz t-norm.

We prove that every finite subset of the integers which tiles by translations satisfies the Coven-Meyerowitz condition T2, with no restriction on the number of prime factors or their exponents. Together with the necessity of T1 and the sufficiency of T1 and T2 proved by Coven and Meyerowitz, this gives their proposed characterization of finite integer tiles. The proof uses strong induction on a cyclic tiling period. Character identities produce periodic Boolean product stripes; integral descent to a coprime quotient and the Frobenius identity force a common orientation. Independent phase shifts then give smaller-period tilings from which the mixed cyclotomic zeros of the original factors can be recovered. A companion Lean formalization verifies the unrestricted T2 necessity statement.

The conformal geometric algebra Cl(4,2) ~ R(8) carries a natural Z_2-grading defined by the dilatation bivector D=\e_4e_0, which satisfies $D^2=1$ and splits the eight-dimensional spinor module into two four-dimensional eigenspaces $S=S_+\oplus S_-$. We prove that the ad_D-eigenvectors of Cl(4,2), those elements with eigenvalue~$\pm1$ under the Lie bracket $[D,\cdot]$, are exhausted by 16 doublets of the form $T^\pm_J=n_\bullet e_J$, where $n_\bullet \in {n_\infty,n_o}$ is a null vector and $e_J$ is a product of Lorentz-sector basis vectors indexed by $J\subseteq\{1,2,3,5\}$. These doublets satisfy universal algebraic identities: $(T^\pm_J)^2=0$ (nilpotency) and $T^+_J\,T^-_J=(-1)^{k(k+1)/2+1}\,\eta_J 2\Pi_+$ (single-projector proportionality), where $k=|J|$, $\eta_J=\prod_{j\in J}\eta_{jj}$ is the spectator metric, and $\Pi_+=\tfrac12(1+D)$. The contracted sums $\sum_{|J|=k}T^+_J T^-_J$ are shown to be controlled by the elementary symmetric polynomials $S_k$ of the Lorentz signature values $(+1,+1,+1,-1)$, with the vanishing of $S_2=0$ reflecting the $(3,1)$ signature of physical spacetime. The even-grade doublets (belonging to Cl^+(4,2) ~ Cl(4,1)) contribute $8\Pi_+$ to the unweighted grand total, while the odd-grade doublets cancel exactly. The 32 doublet eigenvectors generate, under the commutator bracket, the 63-dimensional Lie algebra sl(8,R) in which the conformal algebra so(4,2) embeds as a 15-dimensional subalgebra. All identities, including a closed form for the commutators across the Levi interface, are proved in every conformal extension Cl(p+1,q+1); there the commutator span misses precisely the center of the algebra, a parity dichotomy invisible at the Lorentz signature. Two applications are developed: the top rung of the tower computes the Dirac versus Majorana dichotomy of rank-five pseudoscalars ... [shortened]

We study the algebra of integro-differential operators with smooth coefficients and kernels on a subspace of $C^\infty[a,b]$. We find a normal form for elements of this algebra and determine its unit group. The formulation of inverses gives explicit solutions of inhomogeneous linear Volterra integro-differential equations and Volterra integral equations of first kind with smooth kernels.

We determine the sharp total-support uncertainty for the standard complete set of mutually unbiased bases in every prime dimension at least five, proving that the minimum is \(T_s(p)=p^2-p+2\). This resolves all prime dimensions left open by Fiorentino and Weigert, beginning with dimension eleven, and gives a uniform analytic determination of the optimal lower bound in dimension seven, where the earlier determination was partly numerical. The main new ingredient is a sharp zero-count theorem for arbitrary complex Fourier data supported on the finite parabola. A cyclotomic reduction and truncated local expansion produce a first nonzero finite-field jet that is affine in the transverse variable, reducing the planar zero count to a one-dimensional multiplicity problem. We then classify all extremizing projective states and derive an exact enumeration formula. In dimension seven, this gives analytic four-point extremizers and exactly \(12348\) such rays, completing the analytic description and count left open in the earlier work.

In this paper, we investigate the qualitative properties, long-term dynamic behavior, and stability of a longitudinal mathematical model describing the progression of Type 2 Diabetes Mellitus (T2DM). We significantly extend and refine the mathematical analysis of the model recently proposed by De Gaetano et al. (Journal of Theoretical Biology, 2024). Specifically, we analyze both the autonomous system with a fixed age parameter and the non-autonomous system incorporating explicit time-dependent aging dynamics under generalized conditions, providing rigorous proofs for global existence, uniqueness, positivity, asymptotic bounds, and stability properties.

Let $P$ be a point inside a triangle $T$. We consider the average length of the chords of $T$ through $P$, where the direction of the chord is chosen uniformly. An elementary formula is obtained in terms of the distances from $P$ to the sides and vertices of the triangle. Several classical triangle centers give especially simple specializations. For example, if $I$ is the incenter, then \[ M_T(I)=\frac{2r}{\pi} \log\left(\cot\frac A4\cot\frac B4\cot\frac C4\right). \] Our main result is the sharp inequality \[ M_T(P)\le \frac{p}{\pi\sqrt3}\log(2+\sqrt3), \] valid simultaneously for every triangle of perimeter $p$ and every interior point $P$. Thus, among all such pairs $(T,P)$, the largest possible average chord length occurs only when $T$ is equilateral and $P$ is its center. The proof is an elementary symmetrization argument. We close with brief remarks relating the problem to the radial center of a convex body, the electrostatic potential center of a triangle, and dual quermassintegrals.

We introduce a phase-angle framework for constructing and classifying n-tuple-periodic solutions of the incompressible Navier-Stokes equations, and use it to determine, dimension by dimension, when such solutions exist and why they fail. The velocity field is a difference of two n-fold cyclic trigonometric products, made divergence-free for every phase and every n >= 3 by a telescoping identity. The problem then reduces to two explicit conditions on the n phase angles: a reduction condition, necessary and algebraic, and a pressure-integrability condition, sufficient and differential. Separating them allows a failed construction to be located rather than merely recorded. Unlike the stream-function route on which earlier tri-periodic solutions rest, the framework is not confined to three dimensions, and does not presuppose the Beltrami property. For 3 <= n <= 8 we prove that unforced solutions exist at n=3 and n=4 and nowhere else. In three dimensions the admissible phases form exactly two families, mirror images distinguished by the sign of the helicity; these families are known, and we establish that the construction admits no others, settling a completeness question left open when they were found. In four dimensions integrability reduces a one-parameter curve to a single family, so the quarter-period offset is forced, not assumed. The obstructions divide by parity: arithmetic at n=5,7, where every admissible phase vector annihilates the field; differential at n=6,8, where the field survives but its convective term is not a gradient. Every obstructed dimension nevertheless carries an exact forced solution in closed form. These are not mere self-balancing: the phases are those at which strain and rotation are in exact pointwise balance, so that no pressure is generated at all. What obstructs the unforced problem becomes, in the forced problem, an exactly prescribable input.

Weak formulations for Stieltjes differential equations must retain the directional information carried by atoms of the driving measure. We derive a lateral integration-by-parts identity in which the posterior trace of the test function is forced by the atomic product rule. The resulting dual formulation gives an exact global characterization of integral Stieltjes–Sobolev spaces for the full exponent range from one to infinity and reconstructs the unique strong representative from its traced weak state. We prove that the associated traced weak operator is closed, establish Poincaré–Stieltjes estimates, and \rev{characterize Stieltjes absolute continuity inside bounded variation through a canonical derivative measure whose restriction to the open interval is the classical distributional derivative}. The framework is then applied to finite-dimensional nonlinear measure-driven systems, yielding equivalence of weak, integral, and measure formulations, existence and uniqueness under integrable Lipschitz assumptions, compactness-based existence, and intrinsic reset laws. Cubic and logistic models illustrate the simultaneous encoding of continuous evolution and instantaneous interventions.

We evaluate the six odd class-number-four cases left unevaluated in Ramanathan's treatment of the Rogers-Ramanujan continued fraction. Let \[ S(q)=-R(-q),\qquad R(q)=\cfrac{q^{1/5}}{1+\cfrac{q}{1+\cfrac{q^2}{1+\cfrac{q^3}{1+\cdots}}}}. \] For $n=39,87,111,119,159,287$ we determine the value of $S(e^{-\pi/\sqrt{5n}})$ by applying the genus-character form of the Kronecker limit formula to the four ideal classes of $\mathbb Q(\sqrt{-5n})$. The resulting expressions are given in terms of fundamental units of real quadratic fields. In particular, if $X_n=S(e^{-\pi/\sqrt{5n}})$, then \[ X_n^{-5}+11-X_n^5=\frac{5\sqrt5}{U_n}, \] where the six quantities $U_n$ are displayed explicitly below. We also give the corresponding radical expressions and quartic algebraic certificates, together with numerical checks of the six evaluations.

Let $a,b,c$ be the side lengths of a triangle, and let $R$ and $r$ denote its circumradius and inradius, respectively. Liu proposed the conjecture \[ \sum_{\text{cyc}} \left(\frac{a(b+c-a)}{bc}\right)^k \ge 2+\left(\frac{2r}{R}\right)^k,\qquad k>1, \] with the reverse inequality for $k<1$. We prove this conjecture by reducing it to an algebraic inequality for three positive variables with prescribed sum and product. We also determine the equality cases.

In the paper, the author develops a general convexity lemma and applies it to demonstrate the monotonic decrease of a function involving polygamma functions, arising from the sectional curvature of the Fisher–Rao manifold of beta distributions.

In this note we unveil some subfields of the field of complex numbers which are well known to specialists, but seem to remain hidden for a wider mathematical audience. The existence and construc- tions of these fields follow directly from famous results of the 20-th century, however standard graduate textbooks avoid these topics.

This paper contains a detailed proof of a 1941 conjecture of S.S. Cairns concerning the space of geodesic triangulations of the 2-sphere. The methods are neither combinatorical nor Riemannian but belong to differential topology.

We present generic expressions for one-sided series of Bessel functions of the first kind: $\sum_{m=0}^{\infty} J_{Nm+p}(x)$ and $\sum_{m=0}^{\infty} (-1)^m J_{Nm+p}(x)$. The result is expressed in terms of incomplete Lipschitz-Hankel integrals of Bessel functions of the first kind, which can be represented as Kampé de Fériet functions.