Papers

We construct a bounded complete dcpo P whose maximal point space is homeomorphic to the set R of all reals equipped with the cocountable topology, thereby answering an open problem. Using this dcpo, we obtain a new complete lattice whose Scott space is non-sober.

In this paper, we focus on the analytical description of the interaction between a rarefaction and dispersive shock wave across both convex and non-convex cases, with particular attention to the effect of the non-convexity, within the framework of the Gardner equation. For convex structures, internal oscillations degenerate into small amplitude harmonic waves or a modulated soliton train as t tends to infinity, accompanied by either a retained rarefaction part, or a new dispersive shock wave emanating from it. Taking into account the non-convexity, when alpha>0, we find that kinks either remain non-participating in the interaction at all, or only act to switch polarities of convex structures. As for alpha<0, we solve the Gardner-Whitham equations with three varying Riemann invariants, to analyze the rarefaction-contact dispersive shock interaction where the only possible configuration is that the rarefaction wave is on the left. It is demonstrated that the rarefaction wave will be completely drawn into the interaction region, with a changed contact dispersive shock wave escaping from the left. And internal oscillations eventually degenerate into an asymptotic algebraic soliton train as t tends to infinity. In addition, we study the interaction between a rarefaction wave and composite structure consisting of the contact and classical dispersive shock parts under two distinct situations: (i) For the composite structure-rarefaction interaction, the contact part remains inactive in the interaction, and internal oscillations ultimately degenerate into a contact dispersive shock wave. (ii) For the rarefaction-composite structure case, the entire composite structure participates in the interaction, during

We propose a gravity-consistent arbitrary Lagrangian–Eulerian finite element method for incompressible Navier–Stokes free-boundary problems with moving contact lines. A direct body-force discretization of gravity may fail to ensure consistency between the discrete gravitational work and the variation of the gravitational potential energy on the evolving domain, resulting in an artificial consistency error and persistent spurious velocities near equilibrium. To remove this inconsistency, we reformulate the gravitational potential energy variation as a moving-boundary integral over intermediate ALE configurations and evaluate it exactly using Simpson's quadrature rule. This leads to a mildly nonlinear fully discrete scheme in which the gravitational contribution is exactly consistent with the discrete potential-energy variation. The proposed method preserves volume exactly, satisfies a discrete energy-dissipation law including gravitational potential energy, and under suitable assumptions, drives the discrete velocity to zero in the long-time regime, thereby excluding persistent gravity-induced spurious velocities. Together with the BGN treatment of the free surface and the MDR bulk mesh extension, the scheme maintains accurate interface tracking and good mesh quality near the moving contact line. Numerical experiments in two and three spatial dimensions confirm the theoretical properties.

We introduce and study a finite simple graph of algebraic origin: the Vieta graph on the solution set over $\mathbb{F}_p$ to a symmetric, multivariate equation which is quadratic in each variable. This construction is a broad generalization of the Markoff graph over $\mathbb{F}_p$, extensively studied in the recent literature. We give a systematic approach, partly based on quadratic character sums, to the following basic counting questions: how many vertices does a Vieta graph have, and what is the degree distribution? We focus on explicit counts, addressing the low-dimensional cases in three and four variables.

This document presents a proof of the estimation of a lipschitz parameter by constructing a consistent lower confidence bound through hypothesis test inversion. We prove that as the number of design points grows to infinity, the test correctly identifies any underestimated Lipschitz constant with asymptotic probability 1. The convergence proofs rely on a Uniform Law of Large Numbers and analyze discrepancy metrics under both bounded and unbounded replication sampling.

Distributionally robust optimization (DRO) with optimal transport ambiguity sets is traditionally solved by reformulating the minimax problem into a single-level convex program. While theoretically tractable, these reformulations introduce numerous auxiliary variables and demanding conic constraints that scale poorly in practice. In this paper, we address this challenge by reducing the inner worst-case expectation problem exactly to a scalar budget allocation task. This structural insight yields an efficient algorithm that bypasses large lifted reformulations, alongside a fast post-processing scheme to recover an optimal worst-case distribution supported on at most $N+1$ points, where $N$ denotes the sample size. We embed this procedure within an oracle-based distributional best-response framework to directly compute an approximate primal-dual solution to the overall DRO problem. Furthermore, we extend our analysis to the dual DRO formulation, proving the existence of a least-favorable distribution supported on at most $\min\{N+n+1, KN\}$ atoms, where $n$ and $K$ denote the decision dimension and number of loss components, respectively, and provide an efficient convex programming reduction to extract it from the solution of the primal DRO. Numerical experiments demonstrate that the proposed approach significantly outperforms state-of-the-art reformulation-based solvers.

We develop a quantitative Fredenhagen–Haag like approach for describing Hawking radiation using massless scalar fields on controlled spherically symmetric Vaidya space-times. The local thermal character of the system is supplied by the universal scaling limit of Hadamard two-point functions at an outer trapping horizon \cite{KurpiczPinamontiVerch2021}. On regular detector–horizon windows, we construct globally hyperbolic developments and compare the nonautonomous Vaidya evolution with a frozen Schwarzschild propagator on Sobolev energy spaces. We derive an explicit Duhamel estimate that is uniform in angular momentum, calculate the exact linear null-peeling coefficient, and bound the quadratic remainder of the ray map. Combining these estimates with positivity gives a two-sided detector-response inequality relative to the local thermal reference. Its error terms quantify operator variation, stationary scattering tails, finite Hadamard scaling, horizon localisation, and the outgoing channel. The inequality is valid at finite parameters because each of these contributions is retained. Under the decay hypotheses, the detector response converges to the corresponding Fredenhagen–Haag form for asymptotically stationary accretion and for evaporation–accretion turnaround profiles. We also construct a Hadamard state by Cauchy transport from an eventually stationary Unruh covariance and show that a finite evaporating slab does not determine a late-time response without a prescribed future extension. For asymptotic evaporation with $m(u)>0$ at every finite time and $m(u)\to0$, a mass-rescaled conformal formulation yields a scale-covariant finite-window estimate for scale-following detectors.

This paper presents purely convex programs for passively safe impulsive rendezvous and proximity operations in cislunar orbits. Approach, arrival, and abort maneuvers are all designed and validated in the context of maneuver execution error and navigation uncertainty, and formulated for efficient onboard execution in the autonomous scenario. The outlined methods form the baseline onboard guidance routines for NASA's CAPSTONE 02 mission planned to demonstrate autonomous rendezvous and proximity operations capabilities in the southern 9:2 synodic near rectilinear halo orbit. High fidelity closed loop Monte Carlo simulations using the planned relative navigation sensor suite and measurement cadence verify the intended maneuver design performance.

In this work, we establish new conditional Lipschitz stability results for electrical impedance tomography (EIT) with anisotropies, of recovering the conductivity in a conformal class of a known anisotropic conductivity in both two- and multi-dimensional cases. Then we employ the stability theory to understand the property of the deep Calderón method, one deep learning-based technique for image reconstruction in EIT that has shown promising empirical results, but still lacks theoretical underpinnings. Specifically, we relate the stability theory to the robustness of the method with the proper choice of the training data, and present numerical results in two-dimension to complement the theoretical analysis.

Let $d\ge5$, and let $A_\lambda$ denote the normalized discrete spherical average on $\mathbb{Z}^d$ with squared radius $\lambda$. For a strictly increasing sequence $(\mu_k)$ of positive integers, set $\lambda_k=\mu_k!$ and consider the lacunary maximal operator $A_\star f:=\sup_k|A_{\lambda_k}f|$. For each fixed $\gamma>0$, we prove that the estimate \[ \#\{x:A_\star f(x)>\alpha\} \le K\sum_x\frac{|f(x)|}{\alpha} \left(1+\log^+\frac{|f(x)|}{\alpha}\right)^\gamma \] holds with some finite $K$ for all $\alpha>0$ and finitely supported $f$ if and only if \[ \sup_{N\ge2}\frac{\#\{k:\mu_k\le N\}}{(1+\log N)^\gamma}<\infty. \] The summand is interpreted as zero when $f(x)=0$. The case $\gamma=1$ answers the $\ell\log\ell$ endpoint question of Kesler, Lacey and Mena in the formulation stated above. For $\mu_k=2^{2^k}$, these estimates hold for every fixed $\gamma>0$, but $A_\star$ is not of weak type $(1,1)$. For factorial blocks formed from arbitrary subsets of dyadic grids at widely separated scales, we characterize restricted weak type $(p,p)$, $1<p\le d/(d-1)$, by a uniform covering condition. In particular, for each $1<p_*\le d/(d-1)$, we construct a set of squared radii whose maximal operator is of restricted weak type $(p_*,p_*)$, of strong type $(p,p)$ for $p>p_*$, and unbounded on $\ell^p$ for $1\le p<p_*$.

We give a negative answer to Kourovka Notebook Problem 21.88: no finite group of odd order has commuting probability $1/17$. This follows from a structural theorem asserting that, whenever $p$ is an odd prime and $cp(G)=1/p$, a Sylow $p$-subgroup of $G$ is normal and abelian. Together with Burnside's congruence for the number of conjugacy classes of a group of odd order, this also excludes $cp(G)=1/p$ for every odd prime $p<97$. We further study the next unresolved case not excluded by this congruence, namely $p=97$.

Unit excitation (UE) is a common assumption in stochastic saddle avoidance: the stochastic error must have a uniformly positive component along every direction, in expectation. This condition gives a direct way to rule out convergence to strict saddles, but it also oversimplifies the actual noise structure, and does not match many stochastic optimization regimes. In overparameterized or interpolation models, the noise may vanish near stationarity. In finite-sum problems, the stochastic gradient noise may lie in a low-dimensional, data-dependent subspace. In these (common) scenarios, UE is naturally not satisfied. In this paper, we prove an abstract almost sure avoidance theorem for stochastic recursions without UE. The theorem replaces UE-type requirements by verifiable pathwise conditions. In applications, these conditions follow, e.g., from local smoothness and finite-moment assumptions under standard i.i.d. sampling, or from the finite-sum structure under without-replacement sampling. Since the stochastically sampled maps generally do not share a fixed point, the celebrated center-stable manifold argument used in deterministic analyses is not directly applicable. Instead, we use a path-dependent change of variables together with a pathwise Lyapunov–Perron-based proof strategy. As applications, we obtain strict saddle avoidance for stochastic mirror descent (including SGD) and for random reshuffling. For nonsmooth composite objectives, we prove avoidance results for a proximal-type stochastic gradient method. Combining these insights with suitable iterate convergence guarantees, this allows establishing convergence to local minimizers of the original objective function.

The graph minor project contains the most influential results in recent undirected graph theory research. There has been progress in recent years in generalising some of their results to directed graphs, with the directed grid theorem of Kawarabayashi and Kreutzer [STOC '15] and the directed flat wall theorem of Giannopoulou, Kawarabayashi, Kreutzer, and Kwon [SODA '22]. We discuss the two different versions of the existing directed flat wall theorem and their drawbacks. Then, we present an alternative directed flat wall theorem that excludes a different digraph as a butterfly minor. This new theorem lies “in between” the two existing ones and, as such, does not have either of these drawbacks. The proof of our flat wall theorem is based on the one by Giannopoulou, Kawarabayashi, Kreutzer, and Kwon [SODA '22], which has been adapted by Giannopoulou and Wiederrecht [STOC~'24]. Here we make further adjustments to match our setting.

We study two-times monogenic quartic orders; i.e., those of the shape $\mathbb{Z}[\alpha] = \mathbb{Z}[\beta]$, with algebraic integers $\alpha$ and $\beta$ not $\mathbb{Z}$-equivalent. Two specific types, describing possible algebraic relation among monogenizers of two-times monogenic orders were defined by Bérczes, Evetrse, Győry, who proved under certain conditions on the Galois group of the normal closure of a given number field $K$, that there can be only finitely many two-times monogenic $\mathbb{Z}$-orders in the ring of integers $K$ which are not of these specific two types. In this article, we prove this fact for all quartic number fields.

Higman's conjecture that Z[G] has only trivial units, for G torsion-free, is open for the Promislow (Hantzsche-Wendt) group P, the group over which Gardam disproved the field-coefficient unit conjecture in 2021. We introduce an exact reduction of the integral conjecture for P modulo 2 into two sub-problems, record the base cases as consequences of the Craven-Pappas small-length theorems, and argue that the frontier of the integral problem is word-radius 4: a triviality theorem for Z[P] there would be the first statement separating Z from every field. Throughout, claims are ball-limited and stated as such; we make no claim on the full conjecture.

The topological Tverberg theorem asserts that if $r$ is a prime power then for any continuous map $f\colon \Delta_{(r-1)(d+1)}\rightarrow \mathbb{R}^d$ from the $(r-1)(d+1)$-dimensional simplex $\Delta_{(r-1)(d+1)}$ to $\mathbb{R}^d$ there exist $r$ pairwise disjoint faces of the simplex whose images have non-empty $r$-fold intersection. By refinement, the same conclusion holds if the simplex is replaced by any polytope of the same dimension. While this dimension is tight for simplices, recent work of Soberón and Zerbib shows that this need not be true for polytopes in general. Here we give topological Tverberg theorems for products of simplices. Each of these improves upon the $(r-1)(d+1)$-dimensional threshold, even while imposing the structural condition that the “Tverberg faces” of the product are themselves the products of pairwise disjoint faces from each simplex factor. As before, refinement extends these results, and in particular their dimensional improvements, to products of arbitrary polytopes. As an example, if $d+1$ is a power of two then whenever $m\geq n\geq d+1$ and $m+n=3d+2$ we show that any continuous map $f\colon \Delta_m\times \Delta_n\rightarrow \mathbb{R}^d$ admits disjoint faces $\sigma_1,\sigma_2$ of $\Delta_m$ and $\tau_1,\tau_2$ of $\Delta_n$ such that $\cap_{i,j\in[2]} f(\sigma_i\times\tau_j)\neq \emptyset$. In the case of multilinear maps, our results imply partitions of grid-indexed point sets in $\mathbb{R}^d$ by specialized subsets with stronger intersection conclusions than given by Tverberg's original theorem. Lastly, we extend our results to van Kampen–Flores type theorems which impose dimensional restrictions on the faces of each product factor.

In this paper, we give a memorial tribute to Prof. Erik M. Bollt (1967–2025).

We prove that every finite group that is the product of every pair of its non-conjugate maximal subgroups is soluble, answering Problem 10.34 of the Kourovka Notebook. The almost-simple case was proved by Tikhonenko and Tyutyanov. We treat the remaining minimal-counterexample branch, where the unique minimal normal subgroup has the form S^k, with S nonabelian simple and k >= 2. Two automorphism-stable coordinate subgroup classes satisfying a maximal-supplement criterion produce non-conjugate maximal supplements. If the factorization hypothesis held, their product would impose a p-adic divisibility requirement growing linearly with k, while the quotient contributes only coordinate outer automorphisms and a factor dividing k!. A fixed valuation gap therefore excludes every k >= 2 at once. Suitable subgroup classes are constructed uniformly across the infinite families of finite simple groups using parabolic, torus, and primitive-prime-divisor arguments; stable flag parabolics handle graph fusion, while GAP certificates cover designated finite and sporadic cases. The resulting all-k obstruction provides a reusable mechanism for eliminating direct-power socles in finite-group factorization problems.

We recover a shared constitutive generator from equilibrium locations on weighted graphs. For an even $C^2$ log-coordinate potential with positive curvature, the complete two-site response map determines the generator up to positive scale. A sweep over all positive boundary values at one fixed, possibly unknown, unequal weight ratio gives global identification through a Schröder-Koenigs iteration. We extend this result to a three-vertex chain with two observed free vertices under a sweep condition. For variable-ratio control, we prove finite-target recovery, local Lipschitz stability, and noisy-bisection bounds. Fixed-ratio stability and nonparametric estimation under uncertain controls remain open. In the reciprocal family $J_p$, an exact Gaussian model-selection probability benchmarks synthetic experiments under observation noise and errors in variables. These are identifiability results under controls; no measured data or physical system is analyzed.

In this paper, we introduce and study a new class of mappings termed generalized quadrilateral Ciric-Reich-Rus type mappings, which serve as a four-point analogue of Ciric- Reich-Rus type mappings. We establish the existence and non-uniqueness of fixed points, with at most three fixed points possible for these mappings in complete metric spaces under the condition that the mapping has no periodic points of prime period 2 and 3. We determine that while these mappings are often discontinuous within their domain of definition, they must be continuous at their fixed points. By applying additional constraints, notably asymptotic regularity and continuity, we broaden the applicability of fixed-point theorems to include a broader class of mappings. At the end, we attain fixed point theorems for generalized quadrilateral Ciric- Reich-Rus type mappings in metric spaces that are not necessarily complete. Several illustrative examples are provided to validate the theoretical results and demonstrate the independence of our new class from existing notions.

Let $R$ be a real closed field and let $X_\Sigma$ be a smooth projective split toric variety. For an explicitly chosen basis rigidification $\mathfrak r$ of its Picard grading, we prove that the pair $(\Sigma,\mathfrak r)$ determines the resulting total Chow–Witt ring and give a terminating finite algorithm for all groups, products, twists, and forgetful maps. The ring is identified with an explicit fibre product of the Picard-graded diagonal $\mathbf{I}$-cohomology ring and the integral inverse images of twisted Bockstein kernels over the mod-$2$ Chow ring. Using real cycle classes, we identify the first factor with the cohomology of the real toric variety with all sign local systems. We construct the mod-$2$ cycle map on invariant divisors as explicit deck-transition cocycles and obtain finite signed boundary and product matrices directly from the fan.

Detecting structure in complex networks, especially those arising from physical systems, is a central problem across the sciences. One approach is via rich club analysis, which identifies important vertices using a centrality metric and measures whether those vertices are more tightly interconnected than expected by chance. While informative, this approach captures only pairwise interactions, missing out on higher-order ones known to shape the structure and function of many complex systems. We propose a hyper-rich club pipeline that asks whether central vertices are more tightly interconnected than expected by chance through hyperedges encoding higher-order interactions, which also enables the inclusion of important, often omitted, directional information. We work in a broad class of hypergraphs, which we call general directed hypergraphs, that includes as special cases undirected hypergraphs, head-and-tail directed hypergraphs, and totally ordered hypergraphs (a hypergraph related to directed simplicial complexes from topological data analysis). This unifies several non-equivalent notions of directed hypergraph under one definition. On these hypergraphs we define a hyper-rich club framework whose concrete construction depends on explicit choices the domain scientist fixes according to their research goals. Particular choices recover the existing rich club notions for graphs and undirected hypergraphs, and yield the first such notion for each version of directed hypergraphs. We demonstrate that the pipeline recovers meaningful structure in data by studying networks of very different origins: connectomes, temporal networks of infectious spread, networks of poems, and the XGI hypergraph database, in each case detecting structure the standard graph rich club misses.

In this paper, we introduce a hybrid metric geometry on the space of absolutely continuous probability densities that combines optimal transport (Wasserstein geometry) and log-ratio composition (Aitchison geometry). The hybrid distance $D_\alpha$ is defined through a Benamou–Brenier-type dynamical formulation that couples spatial transport with a centered reaction term preserving total mass.We prove that $D_\alpha$ is a genuine metric and establish comparison estimates with the Wasserstein and Aitchison distances. In particular, we show that the topology induced by $D_\alpha$ is stronger than the narrow topology and weaker than the supremum topology generated by the Wasserstein and Aitchison metrics. We further prove that the metric space is geodesic. Within this framework, we develop the foundations of a gradient flow theory in the sense of Ambrosio–Gigli–Savaré, including the characterization of absolutely continuous curves, metric derivatives, metric slopes, and formal Jordan–Kinderlehrer–Otto schemes. We also investigate hybrid barycenters and their connections with Wasserstein barycenters and Aitchison barycenters. Finally, we discuss several partial differential equations, including logistic diffusion, Allen–Cahn equations with log-ratio constraints, and chemotaxis models with logarithmic growth, as formal gradient flows associated with the hybrid geometry, and compare the proposed framework with the Wasserstein–Fisher–Rao metric.

Let X be a charming space with a Lindelof Sigma kernel Y, and let B be the closure of Y in X. We show that the question whether every charming space is a D-space can be reduced first to the dense-kernel case and then to a closed core. We define Obs_cc(Y,B) as the set of boundary points x in B minus Y such that, for every open neighborhood U of x in X, the intersection of U and Y is not countably compact, and let H be the closure of Obs_cc(Y,B) in B. We prove that the intersection of H with B minus Y is exactly Obs_cc(Y,B), and our main reduction theorem shows that X is a D-space if and only if H is. We also prove the following sufficient condition. If there is a closed set S contained in B minus Y with compact covering number less than the dominating number d, such that every point of B minus Y minus S has an open neighborhood U in X for which the intersection of U and Y is countably compact, then X is a D-space.

Let R be a ring and let Ch(R) be the category of complexes of left R-modules. We consider the C-E (Cartan-Eilenberg) exact structure on Ch(R) and show that it is an efficient exact category. We prove that if R is a left virtually Gorenstein ring, the pair (C-E(GProj),C-E(GInj)) is a C-E-admissible balanced pair. Under suitable additional hypotheses, the converse holds. We also consider the C-E version of Tate cohomology and establish C-E analogues of the Avramov-Martsinkovsky exact sequences and balance results for both Ext and Tor.

We establish the bound for the second-order Hankel determinant $H_{2,2}(F) = A_2 A_4 - A_3^2$ associated with the class $\mathcal{C}_{B}^{\beta}(\mathbb{B})$ of normalized $\beta$-spirallike quasi-convex mappings of type $B$ on the open unit ball $\mathbb{B}$ of a complex Banach space. By utilizing a generalized framework based on a directional slice homogeneous polynomial expansion, we eliminate the standard, restrictive assumption that the mapping is of the form $F(x) = g(x)x$. Under these weaker operational conditions, we parameterize the targeted scalar invariants $A_n$ via the classical Carathéodory functional parameters. A rigorous optimization analysis proves that the established upper bound is strictly sharp for the classical non-spirallike case $\beta = 0$, yielding a maximal value of $1/8$. This sharp bound is verified by constructing explicit multi-dimensional extremal mappings that lift the corresponding single-variable convex profile. Finally, an unresolved open question regarding the exact variational behavior for $\beta \neq 0$ is formulated.

This is a snapshot of a first part on a possibly much longer text on finite geometries, meaning graphs or finite abstract simplicial complexes. In in this first batch we review 12 subjects: Gauss-Bonnet, Poincare-Hopf, Index expectation, Euler's gem, Euler-Poincare,Unimodularity, Brouwer-Lefschetz, Sphere formula, Level sets, Index formula, Quadratic cohomology and Higher characteristic.

Let $\mathbb{K}$ be a non-Archimedean valued field. Let $n \in \mathbb{N}$. For every $(a_j)_{j=1}^n, (b_j)_{j=1}^n \in \mathbb{K}^n$, we show that \begin{align*} \left|\sum_{j=1}^{n}a_jb_j\right|^2\leq \max\left\{\left|\sum_{j=1}^n a_j^2\right|\left|\sum_{k=1}^nb^2_k\right|, \max_{1\leq j<k \leq n}|a_jb_k-a_kb_j|^2\right\}, \end{align*} \begin{align*} \left|\sum_{j=1}^n a_j^2\right|\left|\sum_{k=1}^nb^2_k\right|\leq \max\left\{\left|\sum_{j=1}^n a_jb_j\right|^2, \max_{1\leq j<k \leq n}|a_jb_k-a_kb_j|^2\right\}, \end{align*} \begin{align*} \left|AM(a_j)_{j=1}^n\right|\left|AM(b_j)_{j=1}^n\right|\leq \max\left\{ \left|AM(a_jb_j)_{j=1}^n\right|, \frac{1}{|n|^2}\max_{1\leq j < k \leq n}|a_j-a_k||b_j-b_k|\right\}, \end{align*} \begin{align*} \left|AM(a_jb_j)_{j=1}^n\right|\leq \max\left\{ \left|AM(a_j)_{j=1}^n\right|\left|AM(b_j)_{j=1}^n\right|, \frac{1}{|n|^2}\max_{1\leq j < k \leq n}|a_j-a_k||b_j-b_k|\right\}, \end{align*} where $AM$ denotes the arithmetic mean. First and second are non-Archimedean versions of Cauchy-Schwarz angle-length and third and fourth are Chebyshev arithmetic mean inequalities. Unlike in the Archimedean case, no conditions are required to derive non-Archimedean Chebyshev arithmetic mean inequalities.

Let $\mathbb{Z}^-=\{-1,-2,\dotsc\}$ and $\mathbb{N}=\{1,2,\dotsc\}$. In 2023, Qi and Lim gave two claims for summing the infinite series $\sum_{k=1}^{\infty}\binom{2k}{k}\frac{1}{\alpha+k}\bigl(\frac{\pm1}{4}\bigr)^k$ for $\alpha\in\mathbb{C}\setminus\mathbb{Z}^-$. In this paper, we confirm these claims by expressing the infinite series $\sum_{k=1}^{\infty}\binom{2k}{k}\frac{1}{\alpha+k}\bigl(\frac{z}{4}\bigr)^k$ and the finite series $\sum_{k=1}^{n}\binom{2k}{k}\frac{1}{\alpha+k}\bigl(\frac{z}{4}\bigr)^k$ in terms of the Gauss hypergeometric function ${}_2F_1\bigl[\tfrac12,\alpha;1+\alpha;z\bigr]$ and a generalized hypergeometric function ${}_3F_2$, for $|z|\le1$ and $n\in\mathbb{N}$. Using the Euler integral representation, we derive closed forms, in terms of finite sums of sines and cosines of $\arcsin\sqrt{z}\,$, for ${}_2F_1\bigl[\tfrac12,\tfrac12+n;\tfrac32+n;z\bigr]$, ${}_2F_1\bigl[\tfrac12,1+n;2+n;z\bigr]$, two related ${}_3F_2$ functions, and the incomplete beta functions $B_z\bigl(\frac12+n,\frac12\bigr)$ and $B_z\bigl(1+n,\frac12\bigr)$. By the Pfaff transformation, we obtain closed forms of several further Gauss hypergeometric functions with parameters shifted by $n$. We also give a closed form of $\bigl[(1-z)\frac{\operatorname{d}}{\operatorname{d}z}(1-z)\bigr]^n\frac{\arcsin\sqrt{z}\,}{\sqrt{z(1-z)}\,}$ and a combinatorial identity.

The objective of this paper is to develop an extension of Kummer's second theorem and to establish generalized forms of four classical combinatorial identities—Knuth's old sum (also known as Reed–Dawson's combinatorial identity), Riordan's combinatorial identity, Gould's combinatorial identity, and Touchard's combinatorial identity—using a hypergeometric-series approach. Several new identities also arise as special cases of our main results.