Papers

The dual affine Robinson–Schensted correspondence and the affine matrix-ball construction give two related parametrizations of extended affine permutations. From the stable-window data of the dual correspondence, we introduce a dual weight \(\beta\) and prove that it is consistent with the original pair \((\lambda,N_0)\). We prove that \(\beta\) and the AMBC weight \(\rho\) have identical monodromy along affine Knuth paths. We further give an explicit relation between \(\beta\) and \(\rho\), showing that their difference depends only on the associated tabloids.

This paper investigates the uniform-in-time strong convergence rates of a fully discrete approximation for the stochastic Cahn–Hilliard equation driven by multiplicative noise in spatial dimensions $d\in\{1,2,3\}$. The proposed scheme combines a spectral Galerkin method in space with a backward Euler scheme in time. The main analytical difficulties arise from the state-dependent stochastic perturbation, the absence of a global monotonicity structure for the nonlinear term, and the fourth-order nature of the Cahn–Hilliard operator. In particular, these features make the derivation of uniform $L^{\infty}$-moment estimates highly nontrivial in three dimensions. For the continuous equation, by utilizing the Itô formula to $\|u\|^p$ and introducing the energy functional $\mathcal{E}(u(t))$, we derive the uniform moment boundedness of the solution. At the fully discrete level, we develop discrete energy estimates and close the required high-order moment bounds through an induction argument. Based on these regularity estimates, we deduce uniform-in-time strong convergence rates for the fully discrete scheme. Moreover, we prove the existence and uniqueness of invariant measures for both the exact dynamics and the fully discrete numerical dynamics. Numerical experiments are provided to confirm the theoretical findings.

Let $F=\mathbb{F}_q((u))((t))$ and $H=\mathrm{GL}_n(F)_L\times\mathrm{GL}_n(F)_R$. An ordered tuple $\alpha=(\alpha_1,\dots, \alpha_n) $ defines a cross-$K_2$ multiplier on the doubled Borel. Twisted equivariantization then yields an exact uniformly smooth categorical principal series $2$-representation of $H$. For unitary parameters, the $\mathbb R((X))$-measure gives a positive Hom-valued formal Hermitian inner product on the spherical part. These constructions provide two-dimensional categorical analogues of selected unramified structures.

This paper concerns the maximum number of limit cycles of generalized Abel differential equations $dx/dt = A(t)x^p + B(t)x^q$, where $A$ and $B$ belong to the linear span of a family of functions having the Chebyshev property. Motivated by a recent open problem posed by Huang et al. (Nonlinearity, 2026), we investigate whether this maximum number can be bounded in terms of $p$, $q$, and the structure of the family. Under some natural hypotheses and by means of first- and second-order analyses using Melnikov functions, we provide lower bounds for this maximum number. In contrast to previous work, no specific form for the coefficients is assumed. We then apply these estimates to Abel equations with trigonometric polynomial, polynomial, and hyperbolic coefficients. In the trigonometric polynomial case, we reestablish the results of Álvarez et al. (J. Math. Anal. Appl., 2008) and Huang et al. (SIAM J. Appl. Dyn. Syst., 2020), while in the polynomial case, we improve the classical lower bound given by Lins-Neto.

We study the one-dimensional stochastic partial differential equation \[ d_t X_t(x)=\frac{1}{2}\Delta X_t(x) +b_1\unicode{x1D7D9}_{\{X_t(x)>0\}} +\sqrt{X_t(x)}\dot W(t,x), \] where $b_1>0$, $\dot W$ is space-time white noise, and the initial condition is a nonnegative, compactly supported continuous function. We prove that its unique weak solution has the compact support property. The drift term lies outside the usual regularity assumptions for the Dawson–Girsanov theorem. Instead, the proof is based on a layer decomposition in which the solution is constructed as the monotone limit of sums of super-Brownian motions with random immigration rates determined recursively by the positivity sets of the preceding layers. By comparison, the compact support property extends to a broader class of bounded nonnegative drifts vanishing at the origin. Finally, support-radius estimates yield a comparison of the total mass of $X$ with a squared Bessel process, which allows us to show that $X$ has a positive extinction probability.

We develop an induction theory for relative BiHom-Hopf modules, the special BiHom-Doi–Hopf case associated with the datum $(H,A,H)$ in which the final copy of $H$ carries its regular right $H$-module coalgebra structure. Let $H$ be a monoidal BiHom-Hopf algebra, let $A$ be a right $H$-BiHom-comodule algebra, and let $B=A^{\operatorname{co}H}$. We show that the balanced tensor product defines an induction functor $-\otimes_BA$ left adjoint to the coinvariant functor. If $H$ has a fixed Haar integral, untwisting yields a Haar identity for the BiHom setting and a canonical projection onto coinvariants. These constructions define a hereditary torsion theory with radical $\kappa$ and a torsion-free reflector $Q(M)=M/\kappa(M)$. The modified induction functor $Q(-\otimes_BA)$ then gives an equivalence between right $B$-BiHom-modules and torsion-free relative BiHom-Hopf modules generated by their coinvariants. Equal structure maps recover the corresponding Hom result, while identity structure maps recover the classical relative Hopf-module setting.

In this paper, we present a computer-assisted framework for the rigorous validation of cusp bifurcations of spatially symmetric stationary periodic patterns in parabolic semilinear partial differential equations. Our approach extends to an infinite-dimensional setting the cusp map formulation previously developed in finite dimensions. We formulate the cusp conditions as a zero-finding problem on a Hilbert space of Fourier coefficients, whose non-degenerate solutions correspond to cusp bifurcation points. A key technical ingredient is a careful treatment of the adjoint multiplication operator in the resulting sequence space, which is more involved than in the finite-dimensional case. Starting from a numerically computed approximation, we develop a constructive Newton-Kantorovich argument to prove the existence and local uniqueness of a nearby zero of the cusp map. The non-degeneracy of this solution directly yields the non-vanishing of the cubic normal form coefficient $c$. To complete the verification, we rigorously enclose the spectrum of the linearized operator, confirming that exactly one eigenvalue has zero real part via Gershgorin-type estimates adapted to the symmetry structure of the problem. As a further consequence, the number $k$ of eigenvalues with strictly positive real part at the cusp and the sign of $c$ (both rigorously certified by the framework) together determine the stability structure of the three coexisting solutions inside the cusp region: bistability arises when $k=0$ and $c<0$, monostability when $k=0$ and $c>0$, and no stable solution exists when $k \geq 1$. We apply the method to the Swift–Hohenberg equation and the Gray–Scott system in both one and two spatial dimensions, obtaining rigorous proofs of cusp bifurcations in all four settings.

A $\varphi$-algebra is either a superalgebra or an algebra with involution. In this paper, we study $\operatorname{T}^\varphi$-ideals associated with unital $\varphi$-algebras whose $\varphi$-codimension sequence exhibits cubic polynomial growth. As a consequence, we obtain a complete classification of all $\varphi$-codimension sequences of cubic growth for unital $\varphi$-algebras. Furthermore, we explicitly determine a minimal-degree multilinear generator for every $\operatorname{T}^\varphi$-ideal associated with unital $\varphi$-algebras whose $\varphi$-codimension growth is at most quadratic.

Recent advances have revealed that scattering amplitudes in certain quantum field theories admit a geometric formulation in terms of positive geometries. In this framework, amplitudes are encoded by differential forms whose boundary structure reflects physical principles such as locality, unitarity, and factorization. This thesis gives a self-contained introduction to positive geometries, with particular emphasis on the Amplituhedron, which describes amplitudes in planar maximally supersymmetric Yang–Mills theory through its canonical form. We develop three related directions connecting positivity, volumes, and singularities. First, we study positivity properties of canonical forms through dual volume representations. For polytopes, canonical functions compute volumes of dual polytopes; we extend this picture to nonlinear positive geometries, uncovering non-negative transcendental measures and links with complete monotonicity. Second, we investigate loop-level amplitudes via their singularity structure. Combining Amplituhedron geometry with Landau analysis, we classify leading singularities of the Wilson loop with Lagrangian insertion and constrain the possible singular loci. Third, we examine conjectural relations between Landau singularities, positivity, and cluster algebras. Using momentum-twistor and Grassmannian methods, we identify recursive structures across loop orders, prove several infinite families of cases, and propose a strategy toward the general conjectures. Overall, the thesis develops the perspective that positive geometry provides a unifying language for the analytic and geometric organization of scattering amplitudes, from canonical forms and volumes to the singularities that emerge after loop integration.

Fluid antenna arrays exploit continuous antenna repositioning within a finite aperture to provide geometry diversity beyond grid-constrained port selection. Every displacement, however, changes both the radiation response and the multiport mutual-impedance network, coupling geometry optimization with the source-voltage constraint. This paper develops an electromagnetic-aware (EM-aware) beamforming framework for planar fluid antenna arrays. Phase retrieval converts an amplitude-only shaped-beam specification into an aperture-compatible complex target, and an EM-aware orthogonal matching pursuit (OMP) method selects grid-constrained initial antenna positions. Continuous refinement then alternates exact voltage-constrained current optimization with movement-constrained projected adaptive moment estimation (Adam) updates of all physical antenna positions. Across independently perturbed symmetric dual-beam targets, the proposed method consistently improves the average mainlobe signal-to-noise ratio (SNR) and reduces the peak sidelobe level (PSLL) over a uniform array and discrete port selection.

Fluid antenna systems obtain spatial degrees of freedom by reconfiguring antenna positions within a confined region, a principle that extends to beamforming: shaped beams can be synthesized using far fewer radio-frequency feeds than candidate antenna positions. When the candidates are densely arranged, however, electromagnetic mutual coupling changes the relationship among terminal voltages, induced currents, and radiated fields, so an uncoupled model no longer describes the hardware and may activate an unsuitable set of ports, distorting the synthesized pattern. This paper develops a mutual-coupling-aware framework that converts the desired beam amplitude into a finite-aperture-compatible complex target and models the complete antenna lattice as a coupled multiport network, selecting the active ports and their source voltages through the coupled voltage-to-field response. Inactive candidate ports remain part of the network and carry induced currents, and every compared design is evaluated through the same electromagnetic model under the same source-voltage budget. Numerical results show that the mutual-coupling-aware design improves both the average mainlobe signal-to-noise ratio (SNR) and the peak sidelobe level (PSLL) over coupling-unaware selection and a fixed array, demonstrating that mutual coupling should be exploited in the design itself rather than compensated only in the final evaluation.

Let $(A,\mathfrak{m})$ be a Cohen-Macaulay local ring of dimension $d$ and residue field $k$. Let $M$ be a maximal Cohen-Macaulay $A$-module. Let $e(M)$ be the multiplicity of $M$ and let $\mu(M)$ denote the number of its minimal generators. (1) Assume $A$ is not a complete intersection. If $\text{curv}(M) < \text{curv}(k)$ then we prove that under mild conditions, $e(M) \geq \mu(M)(1 + \text{curv}(k))$. (2) Assume $A$ is a complete intersection. If $\text{cx}(M) < \text{cx}(k)$ then we prove that $e(M) \geq 2\mu(M)$. In both cases we give examples which shows our results are sharp.

We prove that, on a smooth manifold $M$ equipped with a constant rank horizontal distribution $H$ and a smooth inner product $g_H$ on $H$, any horizontal vector field can be written as a linear combination of, at most, $3N$ Lie brackets of horizontal gradient fields of depth-$(1)$, where $N$ is the minimal immersion dimension of $M$, recovering the Riemannian case trivially when $H$ is the entire tangent bundle. When $H$ is also bracket-generating with a uniformly bounded step, we show that any vector field can be written as a linear combination of a uniformly bounded number of iterated Lie brackets of horizontal gradient fields with uniformly bounded depth, both bounds depending solely on the immersion dimension $N$ and the uniform upper bound on the step of $H$. Utilizing this, in conjunction with the Trotter property, we show that, if the manifold is compact, connected, and without boundary, the group generated by flows of horizontal gradient fields is dense in the identity component of the diffeomorphism group. When the manifold is also Riemannian, we show that the same density statement holds for the group generated by diffeomorphic optimal mass transport maps.

Reusable-rocket powered landing under strong aerodynamics couples variable mass, free final time, and bounded aerodynamic controls through nonlinear velocity-frame dynamics. This paper develops a condensed proportional–integral projected-gradient (PIPG) sequential-convex method whose principal contribution is an exact reduced-space inner architecture. Because the problem contains only six terminal hard equalities and no state path constraints, 217 nodal-state variables and 210 trapezoidal dynamics equalities are eliminated from the 31-node convex subproblem, leaving 101 primal variables and six terminal equalities. Row-orthogonal preconditioning, fixed-size matrix–vector products, and nodewise circular-epigraph projections then yield a customized PIPG kernel. Physical consistency of the angle-dependent axial force is maintained by gradually releasing drag sensitivity between the reference squared angle and an epigraph variable $A$, together with a convex tightness term. A pointwise Hamiltonian argument shows that the fully released limiting subproblem admits a tight optimum satisfying $A=\alpha^2+\beta^2$. Deterministic annealing, two-stage inner accuracy, and a rejected-on-failure threefold extrapolation are secondary outer-loop accelerators.

We bound the error in ground-state expectations of local observables caused by truncating the spatial tails of a gapped quantum lattice Hamiltonian. We show that the error is controlled by the interaction strength discarded near each site, rather than by the extensive norm of the omitted Hamiltonian. If the gap remains open along a path that removes the interaction tail, the resulting bounds are uniform in system size and extend, under suitable assumptions, to thermodynamic-limit ground states. The convergence rate reflects the decay of the interaction tail: it is algebraic for power-law interactions, superpolynomial for superpolynomial interactions, and exponential at any strictly smaller rate for exponentially decaying interactions. For superpolynomial interactions, we also derive a direct infinite-volume estimate using automorphic equivalence. The results extend to isolated low-energy sectors and parity-even fermionic systems. For two-body interactions decaying as $r^{-p}$ in $d$ dimensions, we prove an error bound $O(R^{-(p-d)})$ for $p>2d$, where $R$ is the truncation range. We construct a gapped non-translation-invariant example that saturates this scaling, showing that the bound is optimal for the general class considered. Our results quantify when finite-range truncations faithfully reproduce the local physics of gapped long-range systems.

In this paper we discuss the historical role played by Prof. Giovanni Giorgi (1871-1950) in the development of operational methods in mathematical physics. In the literature, the analysis of the scientific contributions of Giorgi is discussed in many historical papers mainly about the $MKS\Omega$-system and its contributions in the field of electrical engineering. Starting from the obituary written by Prof. Dario Graffi (1905-1990), here we analyze in detail the contributions given by Giorgi in mathematical physics, especially in the framework of the studies started by Heaviside (1850-1925) in order to obtain a symbolic representation of the solutions of differential equations emerging in physical problems. \\ Moreover, we underline the little known contribution of Giorgi to the analysis of derivatives of any real order, namely fractional derivatives, according to the present notation. \\ The main aim of this note is to underline the historical relevance of Giorgi in the mathematical foundations of operational methods in relation to the solutions of concrete problems emerging in applied physics.

Let $G$ be a complex reductive algebraic group, the complexification of a compact Lie group $K$. Consider a holomorphic principal $G$-bundle whose total space contains a $K$-invariant Lagrangian submanifold $L$. We develop a method for lifting holomorphic disks from the base with boundary on $L/K$ to the principal $G$-bundle. We show that the Gross–Hacking–Keel–Kontsevich superpotential restricted to a distinguished class of seeds of the basic affine space is obtained by lifting holomorphic disks in the corresponding flag manifold.

We prove various statistical properties (i.e., decay of correlations, central limit theorems with convergence rates, maximal large deviations and almost sure invariance principles) for non-smooth observables in the form of indicators functions in polynomially mixing hyperbolic dynamical systems. Such results were not known (or even expected to hold) for slowly mixing hyperbolic systems, although they are important for applications to physics and other sciences.

Let $G$ be a finite group, and let $\exp(G)$ denote its exponent. The Erdős-Ginzburg-Ziv constant $s(G)$ is the least integer forcing a product-one subsequence of length $\exp(G)$, while the short product-one constant $\eta(G)$ is the least integer forcing a nonempty product-one subsequence of length at most $\exp(G)$. The natural nonabelian extension of a conjecture [W. Gao, On zero-sum subsequences of restricted size II, Discrete Math. 2003] on the Erdős-Ginzburg-Ziv constant in finite abelian groups predicts that $s(G)=\eta(G)+\exp(G)-1.$ We confirm this equality for every finite nonabelian group $G$ having a cyclic subgroup of index $p$, where $p$ is the smallest prime divisor of $|G|$. As further consequences, we determine all generalized Erdős-Ginzburg-Ziv constants $s_{m\exp(G)}(G)$ for this family of groups.

We extend the scaled boundary cubature (SBC) scheme from planar regions to higher-dimensional regions described by oriented boundary patches. The resulting parametrization transforms integrals over compact regions in $\mathbb{R}^d$ into sums of integrals over the boundary-patch parameter domains and a radial coordinate. In three dimensions, this yields a direct volume-integration rule for solids bounded by affine faces, triangular or tensor-product surface patches, B-spline patches, NURBS patches, and combinations of curved and affine boundary representations. For affine polytopes, recursive application of the scaled boundary map yields nested tensor-product rules over simplex sectors; in three dimensions, these reduce to tetrahedral-sector rules that apply equally to convex and nonconvex oriented polyhedra. We also develop transformations for weakly singular integrands. Placing the scaling center at a point singularity exposes the radial power cancelled by the Jacobian, while generalized radial scalings and Gauss–Jacobi quadrature handle fractional powers. A transverse scaled-boundary map provides the analogous construction for affine singular sets, with straight-line examples in three dimensions. Numerical examples verify polynomial exactness on affine polyhedra and a four-dimensional tesseract, rapid convergence on curved B-spline and NURBS solids, and the expected convergence improvements for point and line singularities. Near-boundary singularity tests also identify when additional patch subdivision or patch-parameter transformations are required.

We study reflection-resonance poles $Z_j=E_j-i\Gamma_j$, $\Gamma_j>0$, of a finite one-dimensional disordered sample of length $L$, coupled at one end to a semi-infinite lead, in the regime $L\gg\ell_L\gg k^{-1}$, where $\ell_L$ is the localization length and $k=\sqrt{E}$. The key step is to relate the resonance density to the reflection coefficient at the complex energy $E+i\Gamma$, corresponding to uniform absorption. Exact finite-chain Kac–Rice and Poincaré–Lelong identities reduce pole counting to a finite-length diffusion of the reflected intensity. For the perfect contact transparency the density crosses over from the localization-controlled $\Gamma^{-1}$ law to the broad-resonance $\Gamma^{-2}$ law at $\Gamma_L=k/\ell_L$, in agreement with $L\to\infty$ result of Fyodorov and Meibohm. Finite length cuts off the $\Gamma^{-1}$ regime at $\Gamma_{\rm ultra}=\frac{e^{1-\gamma_{\rm E}}}{2}\Gamma_L e^{-L/\ell_L}$. We derive the ultranarrow resonances crossover shape as an explicit moving front; for contact transparency $\mathcal T<1$ this scale shifts to $\mathcal T\Gamma_{\rm ultra}$. The same reflection process yields the finite-length Wigner time-delay statistics and, in the weak-absorption limit, the Comtet–Texier distribution. We show that at eigenvalues of the corresponding closed Dirichlet sample the Wigner delay is inversely proportional to the squared boundary derivative of the normalized eigenfunction. This quantity also gives the eigenvalue response to displacement of the Dirichlet boundary at the lead-contact end, and hence the force exerted by the eigenmode on that boundary; we obtain its finite-length distribution. Finally, the finite-transparency resonance density obeys the single-channel Moldauer–Simonius sum rule, linking its perfect-coupling divergence to the $\Gamma^{-2}$ tail. Direct lattice and spectral computations test the crossover and front constant.

For a graph $G$, let $\alpha_1(G)$ be the maximum size of an edge set containing at most one edge from every triangle, and let $\tau_1(G)$ be the minimum size of an edge set meeting every triangle. Erdős, Gallai, and Tuza proved that $\alpha_1(G)+\tau_1(G)=\Omega(m^{2/3})$ for every $m$-edge graph and asked for the optimal asymptotic constant. We prove $$\lim_{m\to\infty} \min_{G,\,|E(G)|=m} \frac{\alpha_1(G) + \tau_1(G)}{m^{2/3}} = \frac{3}{2},$$ thereby establishing that the sharp constant is $3/2$ and solving the problem.

For $m\geq 2$, let $c_p(m)$ be the all-dimensional best constant in $$ \left\|\sum_{k=1}^m A_k\right\|_p \leq c_p(m)\left\|\sum_{k=1}^m |A_k|\right\|_p. $$ Tang and Zhang conjectured an explicit formula for every finite $p>1$. We disprove the conjecture with two explicit real $2\times 2$ rank-one matrices at $p=3/2$. The comparison is certified by seven strict rational inequalities and, in particular, places the attained ratio above $207/200$, while the conjectured constant lies below $207/200$. On the positive side, we prove the conjectured sharp bound for every family of rank-at-most-one summands when $2\leq p<\infty$, and classify all equality cases. We also prove the corresponding endpoint statement for $p=\infty$. Finally, for arbitrary complex matrices, we establish the conjectured sharp constant in the case $m=2$, $p=4$.

We complete our work on $\mathsf{GL}_2$ over $\mathbb{Q}$ in the ramified setting for Beyond Endoscopy proposed by Langlands. We prove that the asymptotic formula for each term of the trace formula when summing over $n<X$ with arbitrary smooth test functions at places in $S=\{\infty,q_1,\dots q_r\}$ with $2\in S$, for the standard representation, is $o(X)$. We prove an identity with a variable $X$, called the limit form of the trace formula for $\mathsf{GL}_2$ over $\mathbb{Q}$, directly. The proof uses Arthur's result on the Fourier transform of weighted orbital integrals to rewrite the term involving intertwining operators, and then compares the expansion with the results of the real case due to Arthur-Herb-Sally and Hoffmann, and the nonarchimedean case by direct computation using Arthur's definition.

Let $(M,\omega)$ be a compact Hermitian manifold, and let $\Gamma$ be a symmetric convex cone. We develop a quantitative regularization method for $\Gamma$-admissible functions. As an application, we prove Hölder continuity for every pluripotential solution of complex $m$-Hessian equations whose right-hand sides belong to $L^p$, for $p>\frac{n}{m}$. These results extend to a more general class of complex Hessian equations satisfying the structural condition used by Guo–Phong–Tong.

We propose several zero-point type arguments based on the inevitable zero point(s) of a spectral gap in the quantum spin system phase diagrams in various dimensions. We consider multi-parameter families of Hamiltonian extending the conventional zero-point theorem that includes only one parameter. Analogously to the zero-point theorem, we only impose model-independent transformation relations along the parameter boundary, rather than specifying any low-energy dynamics or response. We further give a series of conjectures, which generalize our statements in a uniform way. Our results give powerful and universal model-independent constraints on the possible relevant operators for critical phenomena in quantum spin models in arbitrary high dimensions.

We study metric completions of conformally singular Riemannian metrics on the interior of a compact manifold with corners. Given boundary-defining functions \(\rho_1,\ldots,\rho_N\) and nonnegative weights \(\alpha_1,\ldots,\alpha_N\), we compare the product metric \(g_\Pi=(\prod_i\rho_i^{-\alpha_i})g_0\) with the sum metric \(g_\Sigma=(\sum_i\rho_i^{-\alpha_i})g_0\). For a boundary point \(p\), let \(I(p)\) be its active index set, \(A(p)=\sum_{i\in I(p)}\alpha_i\), and \(M(p)=\max_{i\in I(p)}\alpha_i\). We prove that \(p\) lies at finite metric distance precisely when \(A(p)<2\) for the product metric and \(M(p)<2\) for the sum metric. Over every accessible boundary point, the active normal directions collapse to a single completion point, yielding a canonical topological description of the completion. The interaction laws produce different boundary-incidence behavior: product weights can make a deeper face inaccessible even when its constituent hypersurfaces are individually accessible, whereas the sum model is governed by the largest active weight. We also determine the induced geometry on accessible open faces: locally the boundary metric is a snowflake with exponent \(1-A_I/2\) in the product case and \(1-M_I/2\) in the sum case, giving Hausdorff dimensions \(m/(1-A_I/2)\) and \(m/(1-M_I/2)\), respectively.

We study the classification of scalar generalized bi-Hamiltonian structures with semisimple leading terms. As an application, we prove the existence of integrable viscous conservation laws for any given viscous central invariant.

Clinical studies generally assume that group-level averages are useful quantities for guiding individual-level decisions in the clinical care of each individual patient. Precision medicine has notably closed the gap towards truly individualized care through highly refined subgrouping. Today, digital health technologies and other modern sources of dense, personal "small data" enable a different approach to treatment individualization—one that seeks to characterize a single person's own recurring health patterns first and foremost, rather than identifying the best subgroup to which they might belong. In this chapter, we review the key concepts underlying n-of-1 studies, single-case designs, and other "multitudinal" approaches for digital health applications, and explore their relationships to other digital health methods. We also share some promising future directions for "esametry", the statistics of the digitized multitudes within each of us.

For a graph \(G\), let \(\gamma(G)\) and \(\rho(G)\) denote its domination number and packing number, respectively. Let \(c_{\mathrm{sub}}\) and \(c_{\mathrm{cub}}\) denote the respective limsups of \(\gamma(G)/\rho(G)\) over connected subcubic and connected cubic graphs as \(\rho(G)\to\infty\). We prove \[ c_{\mathrm{sub}}\geq\frac{13}{6}, \qquad c_{\mathrm{cub}}\geq\frac{17}{8}, \] by constructing two explicit binary branching families. The connected noncubic subcubic graphs \(\widehat B_t^\star\) satisfy \[ |V(\widehat B_t^\star)|=76\cdot2^t-12,\qquad \gamma(\widehat B_t^\star)=26\cdot2^t-4,\qquad \rho(\widehat B_t^\star)=12\cdot2^t-2, \] whereas the connected cubic graphs \(\widehat B_t^\bullet\) satisfy \[ |V(\widehat B_t^\bullet)|=108\cdot2^t-14,\qquad \gamma(\widehat B_t^\bullet)=34\cdot2^t-4,\qquad \rho(\widehat B_t^\bullet)=16\cdot2^t-2. \] The constructions use the same binary connector composition and closing lemma, with different connectors and initial assemblies. As a consequence, both families give unbounded additive violations of \(\gamma(G)\leq2\rho(G)+1\), disproving the proposed inequality even for connected cubic graphs.