Papers

We prove the analogue of Belinskaya's theorem for measure-preserving flows: two free ergodic measure-preserving flows whose $\mathrm{L}^1$ full groups are isomorphic as abstract groups are conjugate up to a scalar time change. This answers a question posed by François Le Maître and the author. We show that whenever two free ergodic flows generate the same orbit equivalence relation and one is contained in the other's $\mathrm{L}^1$ full group, their positive half-orbits are commensurate after possibly reversing time. Katznelson's criterion then yields conjugacy after a scalar time change. The key new ingredient is a commensuration criterion asserting that a measurable subset of the real line whose symmetric differences with its translates have finite average measure over the unit interval is commensurate with exactly one of the empty set, the whole line, and the two half-lines. This criterion and its application were discovered autonomously by a two-agent AI system. The author independently verified the proofs and prepared the final text.

Assume $ G $ is a finite group with order $ |G| = pqrs $, where $ p $, $ q $, $ r $, and $ s $ are distinct prime numbers. We prove that every connected Cayley graph of $ G $ contains a hamiltonian cycle. Our result drops all restrictions of all previously known results on hamiltonian cycles in Cayley graphs of groups of order $pqrs$.

We study spectral-set constants associated with the algebraic numerical range in unital Banach algebras. If $\Gamma_n$ denotes the universal constant for algebraic elements of degree at most $n$, then \[ \Gamma_1=1, \qquad 2n-1\leqslant\Gamma_n<\infty\qquad(n\geqslant2). \] Thus the algebraic numerical range is a spectral set with a constant depending only on the algebraic degree. In degree two we obtain \[ 3\leqslant\Gamma_2 \leqslant\sqrt{1+(2\mathrm e-1)^2}<4.55. \] For unital $C^*$-algebras we prove a gap theorem: the algebra-level constant is one precisely in the commutative case, whereas every non-commutative algebra has constant at least two. Moreover, the constant of the Jiang–Su algebra is exactly the universal Crouzeix constant. In the opposite direction, we construct a norm-one operator with a contractive polynomial calculus on the unit disc but infinite numerical-range spectral constant, and we show that the canonical left shift on every spreading combinatorial space in a broad class has infinite constant. These results settle the three questions posed by Blazhko, Homza, Schwenninger, de Vries, and Wojtylak.

We study a family of mod-eight false-theta moment quotients and determine their radial behaviour at every root of unity. The main structural result establishes an exact finite Fourier closure for the associated periodic boundary sequences. This closure yields a completed functional equation and proves the nonvanishing of the relevant periodic Dirichlet values at all negative odd integers. Consequently, each quotient exhibits either universal normalized factorial asymptotics or a finite nonzero radial value. The finite values belong to cyclotomic fields, satisfy Galois covariance, and specialize at the principal root to signed odd Springer numbers. We also determine exact coefficient divisibility, construct an optimal coefficientwise two-adic interpolation, and identify the unique dominant zero of the common denominator. This zero gives an unconditional coefficient decomposition. Finally, using exact boundary estimates and Möbius inversion, we prove positivity of every exponent in the formal Euler transform of the denominator.

This expository note advertises a little known but very old one-line solution of the Hopf invariant one problem and speculates on the (in)applicability of the idea to the Kervaire invariant one problem.

I present an overview of different phases used in the literature to describe the phase of travelling waves in reaction-diffusion equations. The discussion is tailored to understand the effectiveness of these different phase descriptions in a stochastic setting in 1D. The explicit computation of the autonomous isochronal phase for the Nagumo equation is a new result.

We introduce and investigate the structural properties of super coalition partitions in graphs, a novel direction that bridges cooperative resource deployment with rigid domination criteria. Based on the foundational concept of super domination, a super coalition partition is defined as a vertex set partitioning $\Upsilon = \{A_1, A_2, \ldots, A_k\}$ such that no single class $A_i$ constitutes a valid super dominating set, yet every class can be paired with at least one distinct partner class $A_j$ to form a union $A_i \cup A_j$ that achieves full super domination over the graph. The super coalition number, denoted by $C_s(G)$, represents the maximum possible cardinality of such a partition. In this paper, we establish general operational bounds for $C_s(G)$ using the underlying order and the super domination number $\gamma_{sp}(G)$, demonstrate its relation to the super domatic number $d_{sp}(G)$, analyze its computational complexity proving its NP-complete nature under general conditions, and provide exact determinations for key standard graph architectures including paths, cycles, complete graphs, stars, wheels, and friendship configurations. We conclude by proving that the super coalition number can grow arbitrarily large.

The lottery ticket hypothesis proposes that large random neural networks contain sparse subnetworks that can match the performance of dense models after comparable training. A stronger version asserts that sufficiently overparameterized random networks contain subnetworks that are already accurate before any weight training. Existing theory establishes that such strong lottery tickets exist, but reliable extraction remains difficult. We revisit edge-popup, a frozen-weight score-training method for extracting strong tickets, and identify layerwise sparsity selection as a central bottleneck. We introduce double-scoring, an augmented score-space parameterization that replaces a layerwise sparsity search with optimization over enlarged score tensors. We prove that fixed-density masking in an augmented score space preserves access to all original-coordinate masks, and we show that the resulting method can be interpreted as edge-popup on a zero-augmented network. In controlled experiments, double-scoring substantially improves strong-ticket extraction over fixed-density edge-popup and pruning-at-initialization baselines, improves on the performance of rewound sparse-training topologies, and exhibits markedly lower sensitivity to sparsity hyperparameters. Ablations show that the gain is not merely due to additional trainable score parameters, but is tied to the augmented score-space competition that induces the effective original sparsity.

We consider corner-cutting schemes for closed oriented polygons in the plane. At each step every vertex is replaced by two points on the adjacent edges; the same cutting coefficient is used at all vertices of the current polygon, but this coefficient may vary from step to step. For such schemes we prove the existence of a continuous limit curve and derive an explicit formula for its oriented area. The main result separates two contributions: the geometry of the initial contour enters through its oriented area and a local sum of determinants built from neighboring edges, while the entire dependence on the refinement rule is contained in a single numerical constant. We also discuss the geometric meaning of this constant as an area arising in the standard angle. As applications we consider stationary de Rham curves, the Chaikin scheme, the connection with Archimedes' quadrature of the parabola, and special sequences of coefficients leading to a circle and to an arc of a hyperbola; these give corresponding series for $\pi$ and $\log 2$.

For $a,b,c \in \mathbb{Q}$, the exceptional set associated to the Gauss hypergeometric function $_2F_1(a,b,c;z)$ is defined by $E(a,b,c) := \{ z \in \overline{\mathbb{Q}} \mid {}_2F_1(a,b,c;z) \in \overline{\mathbb{Q}} \}.$ In this paper, the exceptional sets $E(a,b,c)$ are determined explicitly for each $_2F_1(a,b,c;z)$ whose monodromy group is an arithmetic triangle group in Takeuchi's class I. The description is obtained via hypergeometric-modular identities together with transcendence results for periods of abelian varieties due to Wüstholz, and classical result of Schneider on algebraic values of $j$-invariant of elliptic curves with complex multiplication.

Spontaneous stochasticity (SpSt), originating in Richardson's picture of turbulent dispersion and Lorenz's Eulerian view of finite-time loss of predictability, was later formulated under this name by Gawędzki and collaborators and developed in shell models by Mailybaev and collaborators. Whether it occurs in fully developed turbulence remains a major open question. Beyond a few specific classes of systems, however, SpSt has lacked a general mathematical definition. We introduce a measure-theoretic formalism in which it is understood as a measure-selection principle. Given an inviscid problem, a well-posed regularization, and an ambient measure, we study the pushforward of that measure by the regularized flow. Strong SpSt occurs when these pushforward measures converge to a non-Dirac probability law, replacing classical deterministic selection by statistical selection. For finite-dimensional systems, we establish several structural results. Our central attainability theorem shows that, whenever the inviscid problem is nonunique, any probability measure supported on the set of inviscid states can be selected as the limiting law of a suitable regularization. We also identify singular sets in the inviscid dynamics, detected through Dini-type directional growth, as necessary obstructions underlying nonuniqueness. We analyze the relation between SpSt and sensitivity to initial data, clarifying the scope and limitations of turbulence-inspired finite-time separation criteria. Finally, we develop a renormalization-(semi)group viewpoint in which limiting statistics arise as statistical attractors. Explicit examples illustrate how ambient measures, inviscid singularities, regularization scales, and initial-data sensitivity interact in the emergence of SpSt.

We prove that if $A\subseteq[N]$, $|A|\gg\sqrt N$, and $|A_p|\le p/2+O(1)$ for every prime $p\ll \log N$, then $A$ contains at least $\exp\left(c\sqrt{\log N}/\log\log N\right)$ elements in the image of a single integral quadratic $m\pm x^2$. This significantly improves Hanson's logarithmic lower bound, while requiring the residue restriction only for primes $p\ll\log N$. Our proof uses a new weighted entropy argument inspired by our previous work \href{https://arxiv.org/abs/2606.17487}{arXiv:2606.17487} with Sheffer. A matching construction shows that this exponential scale is optimal even when the quadratic may be chosen arbitrarily in $\mathbb{Z}[x]$. We also show extending the residue restrictions to primes up to $(\log N)^3$ yields the stronger bound $\exp(c\sqrt{\log N/\log\log N})$ via a Selberg-sieve-type argument, and discuss an application to the inverse Goldbach problem.

In 2011 G. A. Hiary devised a computational algorithm for the Hardy function $Z(t)$, requiring just $O(t^{1/3} (\log(t))^\kappa)$ operations. This compares to $O(\sqrt t)$ operations necessary for computing $Z(t)$ using the classical Riemann-Siegel formula. The methodology involved the sub-division of the Riemann-Siegel formula into sequences of quadratic Gauss/exponential sums of various lengths $N$. Such sums can be computed rapidly, in order $\log(N)$ operations, using standard recursive schemes. More recently, the principal author developed a similar algorithm with an $O((t/\varepsilon_t)^{1/3} (\log(t))^2)$ operational count, accurate to $\epsilon_t$ in the relative error. Although constructively analogous, the sub-division into quadratic sums was applied to a different asymptotic formula for $Z(t)$, giving the new algorithm an original formulation. This paper presents a significant extension of these ideas. The main theoretical result is an asymptotic expression for $Z(t)$ in terms of sub-sequences of generalised, $m^{\rm th}$-order, Gauss sums of progressively increasing length. Computationally, the main focus falls upon the cubic Gauss sum formulation. The particular parameterisation of these cubic sums makes them amenable to rapid computation, utilising a recursive scheme similar to those implemented for quadratic sums. The net result is a computational algorithm for $Z(t)$ with a reduced $O\bigl((t/\varepsilon_t)^{[.25,.3]}(\log t)^{2+o(1)}\bigr)$ operational count for $t \in [10^{23},10^{35}]$, to high accuracy. Sample computations lend practical support to these findings.

Commuting phase terms in quantum electronic design automation (QEDA) placement circuits are logically invariant under reordering, yet their routed cost varies substantially after hardware mapping, since term order affects CNOT cancellation, interaction locality, and routing pressure. We cast parity/support phase-term ordering within a QEDA phase component as a shielded reinforcement-learning problem: a feasibility shield restricts each step to unemitted terms, so every trajectory is a valid permutation by construction, and an elite (cross-entropy-method) policy is trained against a route-charged proxy combining support-transition size and heavy-hex topology-distance features. We validate by direct Qiskit routing of logically equivalent circuits to a synthetic IBM-style heavy-hex map. On 36-term parity-walk components (50 term seeds x 2 transpiler seeds, statistics at the term-seed level), the per-instance learned ordering reduces mean routed CX to 336.0, a 5.7-12.2% paired reduction over 2-opt and simulated-annealing search at equal or greater proxy budget and 22.3% over the default construction order; routed-CX and routed-depth gains are significant after Bonferroni correction. Honest transfer audits show the proxy is predictive for the parity-walk component but not for extraction-heavy or token/permutation circuits, which require architecture-aware rewards, scoping the contribution accordingly.

We introduce corner rectangle visibility graphs (CRVGs), a combination of two geometrically defined classes of graphs: rectangle visibility graphs (RVGs) and rectangle-of-influence graphs (RIGs). A CRVG has vertices represented by axis-parallel rectangles in the plane, and edges represented by axis-parallel rectangles with one corner at a corner of a vertex-rectangle, an opposite corner at the boundary of another vertex-rectangle, and no vertex-rectangles in their interiors. We also consider CRVGs that only see in one or two directions (south CRVGs and southwest CRVGs). We prove that south CRVGs have at most $\left[\frac{n^2}{4}\right]+n-2$ edges, and this bound is tight. This is the same as the tight edge bound for closed RIGs, but they are different graph classes. We also show that southwest CRVGs have at most $\left[\frac{n^2}{3}+\frac{n}{3}\right]-1$ edges, and this bound is tight. We prove that CRVGs on $n$ vertices have at most $e$ edges, where $\lfloor \frac{3n^2}{8} \rfloor \leq e \leq \lfloor \frac{2n^2}{5} \rfloor$. Finally, we classify several families of graphs as CRVGs, SCRVGs, and SWCRVGs.

We prove a homotopical monadicity theorem for the adjunction between the suspension spectrum and zeroth space functors in motivic stable homotopy theory. Our proof verifies that motivic stable homotopy theory satisfies the hypotheses of the general monadicity theorem of arXiv:2607.12124. In the process, we demonstrate six preliminary results in simplicial motivic homotopy theory, the main technical ingredient being a weak commutativity theorem between the zeroth space functor and realization of simplicial motivic spectra. We also elaborate on a general framework relating monadic algebras under op-lax maps of monads. This monadic framework is used in the proof of the main results and may be of independent interest as well. These simplicial results, and the accompanying monadic framework, may provide tools toward a conjectured operadic recognition principle for motivic infinite loop spaces.

Given only the probability distribution of an aggregate counting variable, what independent latent counting processes are compatible with the observation? Equivalently, when does a probability-generating function admit a factorization into normalized polynomials with nonnegative coefficients? We develop a mathematical theory of such positive factorizations. We introduce the positive factorization poset, whose elements are all positive factorizations ordered by refinement, and define the factorization entropy, measuring the maximal latent Shannon entropy compatible with the observed distribution. We prove a sharp entropy inequality, characterize the equality case by injectivity of the latent addition map, show that entropy optimization may be restricted to maximal atomizations, and exhibit examples where distinct maximal atomizations have different entropy. We further establish support-based obstructions to positive factorization, characterize the real-rooted and Hurwitz-stable sectors, prove a local stability theorem for coprime factorizations, determine exactly the factorable regions in degrees two and three, obtain an exact quartic Hurwitz volume, and investigate the geometry of the factorable region inside the probability simplex through exact calculations and Monte Carlo experiments. These results identify the positive factorization poset as a natural algebraic object associated with probability-generating functions and provide a framework for studying latent independent structure in aggregate counting statistics.

Many signal processing systems ultimately exist to act. Whenever the state variable that determines the action to be taken by a decision maker, or agent, is uncertain, the way that uncertainty is represented decides how well the agent performs and how much its performance can be trusted. This lecture note develops, from first principles and within a single decision-theoretic setting, the link between the {objective} and the knowledge of an agent and the form of uncertainty representation that is sufficient to act optimally. To start, assuming a known environment distribution, we show that a risk-neutral agent needs the posterior distribution over the state, whereas a risk-averse agent can rely without loss of optimality on a {prediction set} and a worst-case decision rule. We then turn to the case in which the environment is unknown, and identify three complementary approaches to address the resulting epistemic uncertainty: calibration of a fixed predictor, credal (ambiguity) sets with distributionally robust optimization, and Bayesian inference over model parameters. The common thread is that reliable decisions require an uncertainty representation matched to the decision objective and to the knowledge profile of the agent, together with a guarantee that certifies the utility the agent will actually obtain.

Pujol and Desfontaines asked whether a private histogram can allow more error on larger counts and use that slack to protect members of larger groups more strongly. We study this question for fixed disjoint groups under add-or-remove-one adjacency. The privacy budget $v(n)$ depends on the affected count, is nonincreasing, and must bound both Rényi-divergence directions at every order. This is the count-dependent form of zero-concentrated differential privacy (zCDP) studied here. The original strict relative-error condition is impossible at count zero. We therefore make the boundary tolerance explicit by requiring $\mathbb{E}\lvert\widehat{x}_i-x_i\rvert < r\max\{x_i,1\}$, without changing the requirement at any positive count. Our main result determines the best dependence on group size. For the upper bound, we directly specialize an existing shifted-transformation framework. The resulting shifted-log Gaussian mechanism has a certified budget $v(n)=O_r(n^{-2})$. Conversely, for every fixed $0<r<1$, any mechanism satisfying the same positive-count utility requirement and count-dependent zCDP must have $v(n)=\Omega_r(n^{-2})$. Thus the inverse-square rate is optimal under the repaired formulation. A many-count information argument further places the leading coefficient in the large-count-then-small-error limit between $\pi/(4e^2)$ and $1/\pi$, a factor below three. At $r=1$, a data-independent release meets the repaired criterion with zero privacy loss.

We prove the reciprocity laws for a family of projective curves over an affine base scheme, where curves can be singular and reducible, and the base can be non-Noetherian. These reciprocity laws generalize the Weil reciprocity law for a projective curve over a field and the reciprocity law for the Contou-Carrère symbol. We apply the proven reciprocity laws to describe the actions of certain central extensions of the group ind-scheme related with the formal punctured disc on certain line bundles on the moduli stacks of quintets. A quintet consists of geometric data including a family of curves and a line bundle on this family. The line bundles on the moduli stacks are constructed using the Deligne pairings of line bundles from quintets.

Embedded boundary summation by parts (SBP) methods define finite differencing based derivative operators with the added feature that the boundary need not coincide with a grid cell, allowing a boundary to be embedded on a regular Cartesian grid. This is achieved by the introduction of interpolation/extrapolation operators that match the accuracy of the boundary closure. These methods have been used to perform black hole excision simulations on a domain with a spherical boundary embedded in a regular Cartesian grid, demonstrating their usefulness for nonlinear problems. In this work, new operators are derived using this embedded boundary framework to increase the order of accuracy of the interior and boundary closure while minimizing the boundary error. Additionally, these novel operators improve the spectral properties on the grid by generalizing to an upwind scheme that has better dispersion relation preserving properties compared to traditional SBP schemes for wave equations. These operators are tested with the curvilinear scalar wave equation on a 3D multiblock grid with an excision sphere embedded in the center block to demonstrate the robustness and accuracy of these novel embedded operators.

We consider a broad family of Heisenberg-type quantum spin systems that were studied by Suzuki and Fisher in 1971. This family includes, for example, the Heisenberg antiferromagnet on any bipartite graph. The ground and Gibbs states of these models are Lee-Yang tensors with radius 1: they are associated with multilinear polynomials that possess an extraordinary zero-freeness property inside the unit polydisk in the complex plane. For each Hamiltonian in this family we show that the spectral gap between the first-excited and ground-state energies is lower bounded by $2\mu$, where $\mu$ is the magnetic field along the $Z$ direction. Using this result we obtain an efficient quantum adiabatic algorithm for the ground energy of any model in this family. The proof is based on a new inequality that relates the spectral gap of a positive semidefinite operator to its Lee-Yang radius – a quantitative strengthening of prior work of the authors that may find applications elsewhere.

A central question in quantum information theory is the circuit complexity of states arising from standard many-body models. We study this question for quantum $p$-spin glasses, random Hamiltonians whose interactions act on $p$-tuples of qubits through Pauli strings. Anschuetz, Gamarnik, and Kiani (arXiv:2404.07231) showed that the optimum energy is separated from the best energy achievable by product states. This leaves open whether shallow circuits can close the gap, since even depth-one circuits can generate entanglement. We show that the entanglement needed to close the product-state gap cannot be generated at shallow depth. When the average interaction degree grows with $n$, we prove that, for all sufficiently large fixed $p$, any circuit preparing an $n$-qubit state whose normalized energy is within a fixed positive constant of the optimum must have depth $\Omega_p(\log n)$. In the bounded-average-degree regime, we prove a fixed-depth obstruction: for every fixed $D$, a sufficiently large degree prefactor rules out depth-$D$ preparation of near-ground states. Both results hold uniformly over circuits with an arbitrary number of ancilla qubits. Our results give an obstruction in the spirit of the No Low-Energy Trivial States problem of Freedman and Hastings (arXiv:1301.1363), but for random quantum spin glasses rather than code-based Hamiltonians such as those of Anshu, Breuckmann, and Nirkhe (arXiv:2206.13228), whose ground states admit polynomial-size preparation circuits. This setting opens a probabilistic route to NLTS-like questions: we recast state-preparation lower bounds for random quantum Hamiltonians as uniform control of Gaussian processes indexed by shallow circuits.

We classify the quasi-Hopf ideals of quasi-Hopf algebras faithfully graded by arbitrary groups. Each ideal determines a normal subgroup, recording which degrees are identified in the quotient, together with an invariant ideal in the corresponding neutral component. We refine this description using an ideal of the original neutral component and a retraction of a naturally associated intermediate quotient. We prove that this intermediate quotient splits as the tensor product of its neutral component with the group algebra of the normal subgroup. For fixed subgroup and neutral-component data, the possible ideals, whenever they exist, form a torsor described by equivariant homomorphisms into central group-like elements. Our results require neither finite dimensionality nor bijectivity of the antipode. We apply them to cocentral abelian extensions with possibly infinite grading, finite-dimensional neutral component, and nontrivial reassociator.

The small Davenport constant $\mathsf{d}(G)$ of a finite group $G$ is the maximal length of a product-one-free sequence over $G$. For the exponent-$p$ Heisenberg group $H_{p^3}$ of order $p^3$, Godara and Sarkar proved $\mathsf{d}(H_{27})=6$ and posed $\mathsf{d}(H_{p^3})=3p-3$ for every odd prime $p$, leaving $p\ge5$ open. We settle the first open case: $\mathsf{d}(H_{125})=12$. The lower bound is the explicit product-one-free sequence $x^4y^4v^4$. For the upper bound we record a product-one criterion that reduces the non-commutative problem to additive combinatorics over $\mathbb{F}_5^2$, and then reduce "every length-13 sequence has a product-one subsequence" to a single finite statement – a spread bound on quotient multisets – which we verify by an exhaustive, memory-flat search in C, its verdict independently reproduced by a second search with a different pruning strategy. Every auxiliary lemma is machine-checked. The argument is genuinely $p$-specific: we identify the exact step that fails for $p\ge7$ (a Chevalley-Warning shortcut whose forced block need not be wide), exhibit the obstructing multiset for $p=7$, and leave only $18\le\mathsf{d}(H_{343})\le24$. The techniques – the Cauchy-Davenport theorem, Chevalley-Warning, and Olson's value of the Davenport constant of $C_p^2$ – are standard; the contribution is their assembly against a new non-abelian target and the finite verification that closes it.

We prove that, for a fixed point in a globally hyperbolic spacetime, the focalization time is locally Lipschitz continuous on the open subset of the future causal cone where it is finite. We also show that the cut time is locally Lipschitz continuous in a neighborhood of any timelike tangent vector whose associated geodesic is defined at least up to (and including) its cut time. Furthermore, we derive quantitative estimates for the Lipschitz constant near the null cone and provide a criterion ensuring that the Lipschitz property extends to null directions. As a consequence, we show that the cut locus of a point has Hausdorff codimension at least $1$. These results extend classical Lipschitz continuity results for complete Riemannian manifolds due to Itoh-Tanaka and Li-Nirenberg. Our approach follows the method of Itoh-Tanaka, suitably adapted to the Lorentzian setting.

Given an $n \times n$ strictly contractive matrix $T$, an (automorphic) Nelson dilation $\widehat{T}$ of $T$ is a certain type of analytic matrix-valued function on the unit disk with $\widehat{T}(0) = T$. Its construction gives a method for lifting a matrix to a matrix-valued function with nice boundary behavior, a trick that has proved useful in recent operator theoretic developments. In this paper, we show that Nelson dilations give a quick way to obtain the minimal isometric and unitary dilations of $T$ and thus, connect naturally to the classical Sz.-Nagy dilation theory. We then initiate the study of the automorphic Nelson dilations as a fundamental object in their own right and prove that every $T$ has Nelson dilations $\widehat{T}$ with particularly useful/interesting properties; for example, they either have strongly entangled eigenvalue functions or have reducing subspaces that are independent of $z$. Along the way, we examine when the product of an invertible matrix and a diagonal matrix has distinct eigenvalues.

We prove that, for $n \ge 5$, an interval group associated with a proper quasi-Coxeter element of the Coxeter group of type $D_n$ admits no surjective homomorphism onto the braid group on $n$ strands. In particular, this provides an alternative proof that such a group is not isomorphic to the Artin group of type $D_n$. The proof relies on techniques from the theory of mapping class groups, and a large part of the paper provides an exposition of this theory for non-specialists.

Given a pair of rational maps $(f, g)$, of degrees $p$ and $q$, each with a parabolic fixed point having a fully invariant simply-connected basin of attraction, we construct an algebraic correspondence $F$ on the Riemann sphere, of bidegree $(pq, pq)$, realizing a mating between the two compositions $g\circ f$ and $f\circ g$ of the maps, and the parabolic faithful discrete representation of the free product of cyclic groups of orders $p + 1$ and $q + 1$. We also show that $F$ is the composition of a pair of deleted covering correspondences of rational maps which are conjugated to polynomials of degrees $p + 1$ and $q + 1$. We generalize our method to construct matings between compositions of pairs of polynomials and (non-parabolic) faithful Kleinian representations of the same group, now with connected regular set. As far as we are aware, these matings between pairs of maps and groups are the first examples that are not time-reversible (that is, they are not conjugate to their own inverses).

We introduce three families of vectors $|\underline{\lambda}^{so}\rangle$, $|\underline{\lambda}^{sp}\rangle$ and $|\underline{\lambda}^{o}\rangle$ parametrized by partitions in the Fock space by using products of adjoint vertex operators. We show that the quotient space of the dual vacuum vector is spanned by the partition vectors indexed by a special family of partitions. The partition-indexed vectors also help us to derive the dual Littlewood identities of types B, C, and D in a new manner associated to the special family of partitions. As an application, we obtain a new free fermionic construction to show that the measures related to dual Littlewood identities introduced by Rains \cite[Section 7]{Rai2000} and Betea \cite[Section 3]{Be2020} are determinantal with repect to some explicit correlation kernels. Furthermore we establish a number of generalized Littlewood identities summed over certain restricted partitions by computing the inner products with elements indexed by one-column partitions {or generalized partitions $(0^m)$} in the complete dual Fock space. {In particular, for each positive integer $n$, we obtain generalized Littlewood identities for $(-n)$-asymmetric partitions. We show that these generalized Littlewood identities contain several well-known Littlewood-type identities as special cases. Consequently we also give a new proof of the generalized Littlewood identity \cite[(5.25)]{LSV2008} for Lie superalgebras. } %We also produce infinite generalized Littlewood identities by calculating the inner products between these elements and some elements indexed by one-column partitions in the complete dual Fock space.