Papers

The paper investigates the topological and metric properties of the set $E$ of values of all infinite continued fractions of the form \[0+\frac{1}{a_1+\dfrac{1}{a_2+_{\ddots}}}=1/a_1+1/a_2+...+1/a_n+...\equiv[0;a_1,a_2,...,a_n,...],\] whose partial quotients ($a_n$) take values in a finite set of positive real numbers $\{e_0,e_1,...,e_{s-1}\}$, $e_0<e_1<...<e_{s-1}$. It is established that the set $E$ is bounded, has cardinality continuum, and is perfect. Conditions under which $E$ is an interval, a nowhere dense set, or a set of Lebesgue measure zero are established. Necessary and sufficient conditions are obtained for $E$ to be an interval and for the corresponding coding system by such continued fractions to have zero redundancy, i.e., for each number to have at most two representations. The geometric meaning of the digits in this representation, as well as the metric relations, is described in terms of the properties of cylindrical sets (cylinders and cylindrical intervals). It is proved that the basic metric ratio, defined as the ratio of the diameter of a cylinder to the diameter of its parent cylinder, is bounded away from both zero and one.

It is shown in this manuscript that a random graph $G$ drawn from the Erdős–Rényi model $\mathcal{G}(n,p)$ with \[ p=p(n)\leq 1/2, \qquad \lim_{n\to+\infty}(np-\log n-\log\log n)=+\infty, \] is a homomorphic core, i.e., every homomorphism from $G$ to itself is an automorphism. This implies tight ETH-based lower bounds of the subgraph isomorphism problem for almost all $k$-vertex patterns with polynomial average degree.

The spectral Turán type problem, initiated by Nikiforov in 2007, aims to determine the graphs among $n$-vertex $H$-free graphs having maximum spectral radius. In this paper, we study this problem for $1$-planar graphs, i.e., graphs that admit a drawing in the plane such that each edge is crossed at most once. Recently, Xu and Chang proved that the graphs among all $n$-vertex $K_5$-free $1$-planar graphs having maximum spectral radius lie within a small family of candidates. First, this paper explicitly identifies the unique spectral extremal graph among the $n$-vertex $K_5$-free $1$-planar graphs. Second, it establishes a structural reduction theorem: For any forbidden subgraph $F$ with $\delta(F)\ge2$ that is contained in $K_2\vee P_{n-2}^{2+}$ but not in $K_2\vee I_{n-2}$, every spectral extremal $F$-free $1$-planar graph contains a spanning complete bipartite graph $K_{2,n-2}$, where $P^{2+}_{n-2}$ is obtained from a path $u_1u_2\dots u_{n-2}$ by adding edge $u_1u_{n-2}$ and all edges $u_iu_{i+2}$ for $1\le i\le n-4$, and $I_{n-2}$ denotes the empty graph on $n-2$ vertices. As applications, the graph among all $n$-vertex $C_5$-free (resp. $2C_5$-free) $1$-planar graphs having maximum spectral radius is determined. These results extend spectral Turán type problems for $1$-planar graphs from cliques to cycles and their disjoint union.

Let $p,q,r\geq 2$ and consider the Campana orbifold \[ \left( \mathbb{P}^1, \left(1-\tfrac1p\right)[0] +\left(1-\tfrac1q\right)[1] +\left(1-\tfrac1r\right)[\infty] \right). \] Primitive positive Campana points on this orbifold correspond to solutions of $a+b=c$ in which $a$, $b$, and $c$ are respectively $p$-full, $q$-full, and $r$-full. We establish upper bounds for the number of such points of bounded height in a broad range of exponents, with a power-saving over the trivial bound. The main analytic input is an estimate for primitive integral points in lopsided boxes on generalized Fermat surfaces $a_1x^p+a_2y^q+a_3z^r=0$, which is uniform in the coefficients.

We study rank-one delta interactions supported by small normal deformations of an \(n\)-dimensional hypersphere \(S_R^n\subset\mathbb{R}^{n+1}\). The interaction is defined by the normalized surface measure on the support and therefore corresponds to the rotationally invariant sector of the hyperspherical shell problem. For the undeformed hypersphere, we derive the bound-state equation for \(E=-\nu^2\) in terms of the modified Bessel product \[ I_{\frac{n-1}{2}}(\nu R)K_{\frac{n-1}{2}}(\nu R). \] We then consider an inward normal deformation \[ X_\varepsilon(\omega)=(R-\varepsilon h(\omega))\omega, \qquad \omega\in S^n, \qquad 0<\varepsilon\ll1. \] Our main result is that, to first order in \(\varepsilon\), the bound-state energy depends only on the average normal displacement \[ \langle h\rangle = \frac1{|S^n|}\int_{S^n}h(\omega)\,d\Omega_n(\omega). \] Equivalently, the deformed hypersphere is spectrally equivalent, up to \(O(\varepsilon^2)\), to a round hypersphere with effective radius \(R-\varepsilon\langle h\rangle\). In particular, mean-zero deformations do not change the rank-one bound-state energy at first order.

The torn-paper channel independently cuts a binary codeword at its internal boundaries and outputs the resulting oriented fragments as an unordered multiset. We consider the critical regime pN log N to alpha, in which the channel capacity is e to alpha. Existing coding schemes use a fixed-density pilot to localize fragments, creating a tradeoff between positional information and payload rate. This paper introduces successive revelation, which partitions the codeword into interleaved tracks and decodes them sequentially. Each recovered payload track becomes an additional positional reference for subsequent stages, allowing progressively shorter fragments to be localized. We establish a finite-track achievable rate whose gap to capacity is O 1 or M for M tracks. Consequently, for every rate below the channel capacity, a finite number of tracks yields a sequence of deterministic codes with vanishing average decoding error probability.

We determine the exact finite-dimensional threshold in the zero-subspace problem of Aron and Rueda for complex homogeneous polynomials. More precisely, for every $d$ and $k$ we determine the least $m$ such that every $d$-homogeneous polynomial on $\mathbb{C}^m$ vanishes on a $k$-dimensional linear subspace. We also determine the exact threshold for arbitrary polynomials of degree at most $d$ to be constant on a $k$-dimensional linear subspace. The two thresholds are different. In the homogeneous case the exact threshold follows from Tevelev's theorem on isotropic subspaces and closedness of the incidence locus. In the bounded-degree case we first eliminate the linear homogeneous component by passing to its kernel; the remaining components, of degrees $2,\ldots,d$, form the system to which the Debarre–Manivel theorem is applied. For $k=2$ we give a separate proof using top Chern classes and Newton's inequalities.

Let $T_n(x)=nx\pmod 1$ on $\T=\mathbb R/\mathbb Z$, and let $\Sigma\subset\mathbb N$ be a finitely generated nonlacunary multiplicative semigroup. We prove that, for every $1\le s\le\infty$, there is an open dense set of potentials $f\in W^{1,s}(\T)$ such that every measure that maximizes or minimizes $\int f\,d\mu$ over the $\Sigma$-invariant probability measures satisfies $h_\mu(T_r)=0$ for all $r\in\Sigma\setminus\{1\}$. For each fixed $s$, a single dense $G_\delta$ subset of $W^{1,s}(\T)$ works simultaneously for all finitely generated nonlacunary multiplicative semigroups. The proof combines Rudolph–Johnson entropy rigidity with periodic-grid perturbations that generically exclude Haar measure from the optimizing faces. In particular, without any uniqueness assumption, the result applies to simultaneous $T_p,T_q$-invariance whenever $p,q\ge2$ are multiplicatively independent.

Hyperbolic surfaces are a fundamental object in mathematics and play an increasingly important role in computational geometry and topology. A key ingredient in the design of efficient algorithms on such surfaces is the availability of a geometric discretization of controlled complexity. In this paper, we present the first algorithm for constructing e-nets on hyperbolic surfaces starting from a fundamental polygon representation. Our approach is based on Delaunay refinement and relies on maintaining Delaunay triangulations through edge flips. The size of an e-net cannot be bounded solely as a function of the genus because of the presence of arbitrarily long collars around short geodesics. To overcome this difficulty, we introduce the notion of a pseudo e-net, which decomposes the surface into e-thin cylinders together with a Delaunay triangulation over an e-net of the remaining thick part. As applications, we obtain algorithms for computing the length spectrum of an e-thick hyperbolic surface and for computing the systole from a pseudo log(sqrt(2))-net. These results demonstrate that Delaunay-based discretizations provide a practical and versatile framework for algorithmic computations on hyperbolic surfaces.

The topological Hedetniemi conjecture asserts that the topological index of the product of two \(\mathbb{Z}/2\)-complexes is equal to the minimum of their respective indices. Bui and Daneshpajouh studied a natural generalization of this conjecture to \(G\)-spaces and proved that this generalized conjecture fails whenever the group \(G\) is neither cyclic of prime-power order nor generalized quaternion. More precisely, for any such group \(G\), they constructed finite free \(G\)-simplicial complexes, each having topological index one, whose product has topological index zero. The purpose of this note is to demonstrate that this discrepancy can be made arbitrarily large. Indeed, for every integer \(n\geq 1\), we show that there exist two finite free \(G\)-simplicial complexes, each of topological index $n$, whose product has topological index zero, provided that \(G\) is neither cyclic of prime-power order nor generalized quaternion. In particular, the corresponding weak form of the generalized topological Hedetniemi conjecture fails in the strongest possible sense for those groups.

We study an operator $\Theta$ on finite integer sequences, where $\Theta(\sigma)_i$ counts the entries to the left of $\sigma_i$ that are strictly smaller than $\sigma_i$. This operator is a variant of the so-called Lehmer code. For every sequence $\sigma$, the image $\Theta(\sigma)$ is an inversion sequence, and the restriction of $\Theta$ to permutations of $[0,n-1]$ is a bijection onto inversion sequences of length $n$. We characterize the fixed points of $\Theta$ by avoidance of the pattern $101$ together with a saturation condition, prove that they are counted by the Catalan numbers, and give an explicit recursive bijection with Dyck paths. We also show that the sequences whose first $\Theta$-image is fixed are precisely those avoiding both $101$ and $201$. Finally, we prove finite stabilization for all inversion sequences, exhibit a family attaining the maximal stabilization time, and show that the second stabilization level is not closed under classical patterns.

We provide an explicit bound for the image of the Zilber–Pink locus of a curve in an abelian variety under reduction modulo a suitably large prime $p$. We also provide a new proof of finiteness of the Zilber–Pink locus. Under further assumptions on the abelian variety, we prove a bound on the size of the locus.

This paper describes the electricity markets and operational processes in the Irish power system and translates them into a Mixed Integer Linear Programming (MILP) model. The model is designed to estimate renewable generation Curtailment and Constraint. A full mathematical formulation is presented and tested on both a simple example and a realistic model of the Irish transmission network. The results show that the proposed formulation works as intended, capturing curtailment and constraint effects, while also highlighting model limitations and directions for future improvement.

In this paper, we study the finite-dimensional, homogeneous, all-to-all coupled Kuramoto model. We begin by performing a complete spectral analysis of all equilibria of the system. Motivated by this analysis, we then derive an explicit description of the unstable manifolds associated with the family of incoherent equilibria. Subsequently, we establish the convergence of this family of unstable manifolds to the Ott-Antonsen manifold $\mathcal{M}_{\mathrm{OA}}$, with respect to the Hausdorff distance induced by the $p$-Wasserstein metric. We further carry out an analogous analysis for the corresponding counterpart of $\mathcal{M}_{\mathrm{OA}}$ in the continuum limit. Moreover, we establish the uniform-in-time convergence of trajectories of the finite-dimensional Kuramoto model on these invariant manifolds towards their corresponding mean-field limit trajectories. Our results provide a direct geometric link between finite-dimensional particle systems and their mean-field, or continuum, limits.

We establish general upper bounds on the maximum effective dimension and information gain of kernel Gram matrices over arbitrary designs in terms of approximation properties of the kernel or, equivalently, of the corresponding reproducing kernel Hilbert space. We show that across three regularity regimes, covering as special cases the Matérn and squared exponential kernels, these bounds cannot be improved beyond constant factors.

We show that the key optimization results of the classical Markowitz portfolio selection theory, originally formulated for variance as the risk measure, remain available in explicit closed form under a broader class of strictly convex quadratic risk measures. The proposed framework replaces the covariance matrix with an arbitrary symmetric positive definite matrix and allows additional linear and constant terms, thereby containing various models arising in transaction cost optimization, benchmark relative optimization, covariance regularization, and factor models. Closed-form formulas are obtained for the efficient frontier, the global minimum risk portfolio, the maximum Sharpe ratio portfolio, the Capital Market Curve, the tangency portfolio, and the maximum utility portfolio. In contrast to the classical Markowitz model, the tangency portfolio does not coincide with the maximum Sharpe ratio portfolio, revealing a new geometric phenomenon. A numerical example confirms the derived formulas.

Let $\Omega\subset\mathbb{R}^d$ be a bounded smooth domain and let $-\Delta_D$ be the positive Dirichlet Laplacian on $L^2(\Omega)$. For a real polynomial $p$ with positive leading coefficient, we study the constrained linear equation \[ u_t=-\Pi_u\,p(-\Delta_D)u, \qquad \lVert u(0)\rVert_{L^2}=1. \] Its solution is the normalized semigroup orbit \[ u(t)=\frac{e^{-t p(-\Delta_D)}u_0} {\lVert e^{-t p(-\Delta_D)}u_0\rVert_{L^2}}. \] The active spectral support is preserved, and the trajectory converges to the normalized projection of $u_0$ onto the active eigenspaces for which $p(\lambda_j)$ is minimal. The next active polynomial spectral value gives the exponential rate. For every $\theta\geq 0$ and $\tau>0$, the same rate holds in the domain of $(-\Delta_D)^\theta$ for $t\geq\tau$, even when the initial datum has no fractional regularity. We also show that a finite set of Dirichlet levels can be prescribed as the global minimizing set of a polynomial, and that an isolated selected set is stable under sufficiently small polynomial perturbations. For $p(s)=s^m$, the lowest active Dirichlet level is selected. For $p(s)=(s-\rho)^2$, selection is by distance from $\rho$, and cross-level degeneracy occurs only at Dirichlet midpoints.

The spectral Erdős–Ko–Rado problem asks for the largest adjacency-tensor spectral radius of a $t$-intersecting $k$-uniform family. Keevash, Lenz and Mubayi proved that, for fixed $k,t$ and sufficiently large $n$, the unique extremal family is a full $t$-star, and asked whether such a theorem extends to all $n$. Let $\mathcal{A}_r=\{F\in\binom{[n]}k:|F\cap[t+2r]|\ge t+r\}$ be the Frankl families and write $\rho_r$ for their spectral radii. For $2\le t<k$ and $n>2k-t$, we prove that $\mathcal{A}_0$ is spectrally extremal if and only if $\rho_0\ge\rho_1$; it is unique up to permutation when the inequality is strict, whereas $\mathcal{A}_0$ and $\mathcal{A}_1$ are both extremal at equality. The layerwise pull used in the Ahlswede–Khachatrian cardinality proof is not applicable here: applied directly, it may decrease the spectral radius. Our proof instead pulls all boundary layers simultaneously and applies Perron tail symmetrization. It follows that $\mathcal{A}_0$ is uniquely extremal for $n\ge (t+1)(k-t+1)+\lceil(t+1)\log(t+1)\rceil$; the leading coefficient $t+1$ is best possible for fixed $t$. We also determine all extremal structures for $t=1$ throughout the range $n\ge2k$.

Starting from an arbitrary conformal net, we construct a genus zero functorial conformal field theory whose Hilbert space is the vacuum sector of the net. Specifically, we construct an algebra for the operad of little discs and conformal embeddings with values in the category of Hilbert spaces and bounded linear maps. We work with closed discs, and our conformal embeddings explicitly allow the image of an incoming disc to overlap with the boundary of the outgoing disc. As an application, we show that all conformal nets satisfy the trace class condition: if $L_0$ is the conformal Hamiltonian of the net, then the operators $r^{L_0}$ are trace class whenever $0 \le r < 1$. In particular, the $L_0$-eigenspaces of a conformal net are automatically finite-dimensional.

Using monthly Niño-3.4 anomalies through July 2026, we investigate how much predictive information is contained in delayed observations of the index. Ridge regression identifies informative delays, while multilayer perceptron and sparse identification of nonlinear dynamics (SINDy) models test whether nonlinear complexity provides additional direct forecast skill; gated recurrent unit (GRU) and long short-term memory (LSTM) networks provide a complementary test in which the temporal representation is learned internally. Delayed observations substantially improve forecasts over persistence and climatology at leads of up to six months, but increasing model complexity provides no systematic improvement. Historical recursive experiments favor a simple explicit SINDy recurrence and select shallow recurrent architectures, with no appreciable gain from learning the temporal representation internally. These results support a compact predictive representation of Niño-3.4 evolution in which the representation of past information is more consequential than model complexity. As a prospective application, the selected models are used to forecast the developing 2026 event beyond the last available observation and to compare its predicted evolution with completed historical El Niño events.

We give a complete rigorous asymptotic analysis of the generating functions of 3-noncrossing skeleton matchings and canonical 3-noncrossing skeleton diagrams. Let $F_3$ be the ordinary generating function of 3-noncrossing matchings, and let $S(y)=\sum_{n\geq 0}S(n)y^n$ be determined by $S(zF_3(z)^2)=F_3(z)$. The proof is deliberately ordered to avoid circularity. First, Lagrange inversion, a Stieltjes representation of $F_3$, exact cut-boundary estimates, and a moving horizontal Hankel contour give $S(n)\sim 24(\pi A^5)^{-1}\sigma^{-n}n^{-5}$ independently of any $\Delta$-analyticity of $S$. This estimate supplies boundary regularity of $S$ and $S'$. We then prove a global biholomorphic inversion theorem, continuation across every nonprincipal point of the convergence circle, and a logarithmically perturbed sectorial inverse theorem. A complete disk-chain and monodromy argument yields a single-valued continuation to a standard $\Delta$-domain. At the principal singularity, $S(y)=Q_4(u)-(\pi A^5)^{-1}u^4\log u+O(u^5(1+|\log u|))$, where $u=1-y/\sigma$. Finally, the canonical composition $S_3^{[4]}(z)=(1-z)(S(\vartheta(z))-1-\vartheta(z))$ is shown to be $\Delta$-analytic at its unique dominant singularity $\eta=0.49340718057613087519\ldots$, and $[z^n]S_3^{[4]}(z)\sim 7892.16205625817\ldots n^{-5}\eta^{-n}$. The argument retains the methods and detailed estimates of the original proofs while closing the analytic gaps in the earlier dissertation treatment.

We prove that every affine terminal Gorenstein toric variety of dimension at most four is unobstructed. In dimension four, the obstruction space need not vanish; instead, we determine the possible homogeneous obstruction degrees and construct simultaneous two-parameter deformations which eliminate the remaining potential obstructions.

In this paper, we establish a connection between joint moments of characteristic polynomials and their derivatives in the Circular Jacobi Ensemble, a generalisation of the Circular Unitary Ensemble, and solutions of nonlinear Painlevé equations. For finite $N$, we show that these joint moments are characterised by a solution of the $\sigma$-Painlevé V equation for all real moment exponents in their admissible range. Under an appropriate large-$N$ scaling limit, we further prove that the limiting joint moments admit a representation in terms of a solution of the $\sigma$-Painlevé III$'$ equation for a certain range of moment exponents. As applications, we answer a question posed by Assiotis et al. in [Math. Physics. Anal. Geom. 25 (2022), no.2, Paper No. 15, 24pp, Remark 1.7] by showing that the characteristic function of a distinguished random variable is connected with the $\sigma$-Painlevé III$'$ equation for complex parameters. Furthermore, we investigate joint moments involving higher-order derivatives of characteristic polynomials in the Circular Jacobi Ensemble. As a consequence, we extend a result of Assiotis et al. in [Comm. Pure Appl. Math. 79 (2026), no. 7, 1771-1827, Theorem 1.11] concerning the joint moments of a sequence of random variables arising from the ergodic decomposition of Hua-Pickrell measures, from real parameters to complex parameters.

We study the center of the quantum Schur superalgebra $\mathcal{S}_v(m|n,r)$ associated with the general linear Lie superalgebra $\mathfrak{gl}_{m|n}$. Using the super Schur–Weyl duality due to Mitsuhashi between the quantum supergroup $U_v(\mathfrak{gl}_{m|n})$ and the Hecke algebra $\mathcal{H}_v(\mathfrak{S}_r)$, we transfer two known bases of the center of $\mathcal{H}_v(\mathfrak{S}_r)$, namely the Geck–Rouquier basis and the Jones basis, to the center of $\mathcal{S}_v(m|n,r)$. This yields two distinct bases for $\mathscr{Z}(\mathcal{S}_v(m|n,r))$, indexed respectively by the symmetrized hook set $H^{\vee}(m|n,r):=H(\min(m,n)\mid \max(m,n),r)$ and by the full $(m|n)$-hook set $H(m|n,r)$. Our approach relies on a detailed analysis of the ring $\Lambda_{m|n}$ of doubly symmetric polynomials satisfying $f|_{x_m=t=-y_n}$ independent of $t$, and of its power-sum bases.

In this paper we study nonlocal nonlinear equations of $s$-fractional $p$-Laplacian type in open subsets of $\mathbf{R}^n$. We investigate how the boundary regularity of solutions depends on the parameters $s$ and $p$, including the local case $s=1$. Specifically, we show exactly when regularity for $(s_1, p_1)$ implies regularity for $(s_2, p_2)$. The proof relies on the equivalence between Wiener criteria formulated with condenser and Sobolev capacities. To establish this equivalence, we derive precise comparison estimates between the two capacities. The Wiener integral defines thinness and the fine topology. We show that every superharmonic function associated with a nonlocal nonlinear operator is finely continuous. Moreover, we prove that polar sets in this fractional setting coincide with sets of zero capacity.

We study a deterministic first-order mean field game model for human capital accumulation in which a continuum of agents invest in education and training to reach a certain qualification level in minimum time. The productivity of learning depends on both the individual's current level of human capital and the distribution of the population, modeling knowledge spillovers and social interactions. Learning is financed by the income generated by existing human capital, leading to a state constraint requiring the financial resources to remain nonnegative throughout the evolution. For a fixed population flow, we formulate the individual problem as a state-constrained minimum-time optimal control problem with time-dependent dynamics. We establish the existence of optimal controls and trajectories, analyze the associated value function through the Dynamic Programming Principle and a Hamilton–Jacobi equation, and characterize the optimal learning effort under suitable regularity and convexity assumptions. The optimal feedback induces a continuity equation describing the evolution of the population distribution. The resulting mean field game is formulated as a fixed-point problem for the population flow. By constructing a compact invariant set and applying Schauder's fixed-point theorem, we prove the existence of at least one equilibrium on every finite population horizon, while the individual minimum-time problem remains formulated on the infinite time horizon.

Kwasik and Schultz constructed two-ended open $4$-manifolds which satisfy the usual finiteness and stability conditions at infinity but do not admit arbitrarily small $1$-neighborhoods. In particular, neither end is collarable, so the manifolds are not completable. We show that the open manifolds associated to their non-desuspendable $C_2$-actions nevertheless admit finite-dimensional compact ANR $\mathcal{Z}$-compactifications. Consequently, there exists a $\mathcal{Z}$-compactifiable open $4$-manifold which is not pseudo-collarable. This answers a question of Guilbault and Tinsley.

This paper utilizes an extremely simple idea: in the multiplicative group of a finite field $F_q$, cosets of a certain subgroup are considered, and an attempt is made to combine these cosets into pairs such that the resulting induced subgraph in the corresponding Paley graph is strongly regular. It is shown that there exists an infinite sequence of Paley graphs that can be partitioned into Petersen graphs in this manner. Furthermore, a new strongly regular graph with parameters $(50, 21, 8, 9)$ is constructed using this method.

We prove the rank-one Lyapunov spectral identity recorded as Problem 9.1 in the 2004 collection of unsolved problems in mathematical systems and control theory. Let $P,Q,R$ solve the coupled discrete Lyapunov and Sylvester equations associated with $A$ and its rank-one update $A_2=A+vw^\top$. When the displayed inverses exist, we show that $P^{-1}RQ^{-1}R^\top$ and $(I+PQ)^{-1}$ have the same characteristic polynomial. A rank-one determinant factorization of the equation for $Q$ produces a scalar bilinear kernel. Evaluating it at the eigenvalues of $A$ and at their reciprocals gives $RQ^{-1}R^\top=BQ^{-1}B=P-BPB$, after which the two target matrices are the same two factors in opposite order. Polynomial continuation extends the identity from a nonempty open set of admissible systems to the full admissible domain and yields a determinant corollary without stability assumptions; when the spectra of $A$ and $A_2$ are disjoint, $Z=b(A)^{-1}P$ gives an explicit similarity. In the Schur-stable realization setting, the result recovers the associated principal-angle and past/future canonical-correlation spectra.

We study distributed hypothesis testing and establish the exact error exponent in single-letter form for new testing problems. In distributed hypothesis testing, a receiver decides between $\mathcal{H}_0:P_{XY}$ and $\mathcal{H}_1:Q_{XY}$ based on $Y^n$ and a rate-limited description of $X^n$. So far, such single-letter forms are known only for testing against independence, studied by Ahlswede and Csiszár, and testing against conditional independence, studied by Rahman and Wagner. In this paper, we study testing against dependence, where $P_{XY}=P_XP_Y$, and show that its error exponent is given by Han's exponent, which is established by single-letterizing a multi-letter version of Han's exponent. We also disprove a previous conjecture of Han claiming that the error exponent is given by the lautum information, as we show that Han's exponent can strictly exceed the latter. We then consider the Cartesian product of testing against dependence and testing against independence, and derive a single-letter characterization of its error exponent. We next study testing against conditional dependence, which is the dependence-testing counterpart of the setting studied by Rahman and Wagner. We derive a single-letter converse bound for the error exponent by introducing and solving a related setting where the side information is also available to the transmitter. We show that our converse bound is tight in some cases by using a conditional-coding version of the quantization scheme. Finally, we extend our converse method for testing against dependence to the general setting, and recover a recently established single-letter converse bound.