The self-averaging of the replica overlap is proven in the Edwards-Anderson (EA) model under random field almost everywhere in the coupling constant space in any dimension. The EA order parameter is represented in terms of the derivative of the free energy density with respect to the random field strength, regardless of boundary conditions. Tasaki's correlation inequality for finite-dimensional spin glass models shows that the expectation of the squared replica overlap is bounded by the squared EA order parameter. These simple evaluations enable us to prove that the variance of the replica overlap vanishes in the infinite-volume limit. The self-averaging of the replica bond overlap is proven also in the EA model with Gaussian exchange interaction without random field. Short-range spin glass models have been shown to behave differently from mean-field spin glass models with RSB phase.

Organizations and researchers show increasing interest in using large language models (LLMs) in place of human participants in A/B tests, in the hope of experimenting faster and at lower cost. We study when a treatment effect estimated on LLM outcomes can recover the effect that would have been measured on the human population of interest. Distributional equivalence between LLM and human outcomes would make any standard estimator valid but is unrealistic. We therefore develop a statistical framework that adapts surrogate endpoint theory to LLMs, showing that calibrating LLM outcomes to human outcomes identifies the average treatment effect under surrogacy and comparability conditions that are jointly weaker than distributional equivalence. We present a falsification test for surrogacy and a bound on the worst-case bias from limited overlap between the LLM and human samples. We further show that the stochasticity inherent to LLMs can weaken surrogacy for identification while also introducing bias and variance during estimation, but that using an average over multiple LLM draws per unit as the surrogate mitigates these issues. Simulations validate the results, and an empirical application to A/B tests on Upworthy headlines shows that raw LLM predictions recover only 39\% of the human treatment effect while nonparametric calibration closes the gap. A central takeaway is that A/B testing on LLMs yields correct results only by assumption, whereas A/B testing on humans is correct by design, and that the required assumptions are hardest to justify precisely where A/B testing on LLMs promises the greatest benefit. We discuss the role of LLM choice, prompting, and temperature as design variables, the compounded challenge posed by long-term outcomes, and how to size human pilot studies for validation.

In this paper, we mainly investigate zero-sum problems over $(\mathbb Z/n\mathbb Z)^r$ (with $n>1$ and $r>0$) of two new types. Let $s_r(n)$ (resp. $t_r(n)$) be the least positive integer $k$ such that for any ${\bf a}_1,\ldots,{\bf a}_k\in\mathbb Z^r$ not congruent to ${\bf 0}=(0,\ldots,0)$ modulo $n$ (resp., with all the coordinates relatively prime to $n$), there is an $I\subseteq\{1,\ldots,k\}$ with $|I|=n$ for which $\sum_{i\in I}{\bf a}_i\equiv{\bf 0}\pmod n$ but $\sum_{i\in I}{\bf a}_i\not\equiv{\bf 0}\pmod {n^2}$. We study lower and upper bounds for $s_r(n)$ and $t_r(n)$. For $n>2$, we conjecture that $$s_1(n)=2n+1, \ t_1(n)=2n-(-1)^n,\ s_2(n)=4n+1,$$ and $$t_2(n)=\begin{cases}4n-3&\text{if}\ 2\nmid n,\\3n-3&\text{if}\ 2\mid n. \end{cases}$$.

We prove approximate tensorization for hypercontractivity and logarithmic-Sobolev constants for a class of reversible quantum Markov semigroups satisfying the positive off-diagonal scaling (PODS) condition. This class includes qubit examples and generalized depolarizing semigroups with respect to full-rank states in arbitrary finite dimensions. For any such semigroup \((Φ_t)_{t\ge 0}\) and every tensor power \(n\), we show that the log-Sobolev constant of the product semigroup \(Φ_t^{\otimes n}\) is at least \(2/(3\ln 2)\approx 0.96\) times the log-Sobolev constant of the single-site semigroup \(Φ_t\), independently of \(n\) and the local dimension \(d\). The proof first establishes an exact tensorization of the \((q,2)\)-hypercontractive inequality for integer \(q\), in particular \(q=3\), and then extends the estimate to all real \(q>2\) by complex interpolation; the standard implication from hypercontractivity to logarithmic-Sobolev inequalities yields the stated almost tensorization result. As an application of the same method, we also obtain sharp \((q,2)\)-hypercontractivity estimates for qubit depolarizing channels.

In this paper, we study the Bloch and $\mathcal Q_p$--Carleson measure problems on the unit disc $\mathbb D$. In the Bloch case, for a positive Borel measure $μ$ on $\mathbb D$, we give a complete characterization of the boundedness and compactness of the embedding $$ \operatorname{id}:\mathcal B \longrightarrow L^2(μ) $$ in terms of the Bloch capacity $\mathfrak B_{\mathcal R}(μ)$ associated with an admissible dyadic resolution $\mathcal R$ of $\mathbb D$. The proof is based on the Bergman projection representation of Bloch functions, conditional expectations on admissible dyadic resolutions, and a finite-dimensional semidefinite programming argument. We also adapt this dyadic framework to the more general $\mathcal Q_p$--Carleson measure problem and obtain a corresponding complete boundedness and compactness characterization for $$ \operatorname{id}:\mathcal Q_p \longrightarrow L^2(μ), \qquad 0<p\le1. $$ This work further develops the dyadic approach introduced in our recent work on composition operators on $\mathcal Q_p$ spaces, but in a different setting where the embedding involves recovering function values from derivative information.

Deep neural networks (DNNs) have achieved remarkable success in scientific computing, yet they often suffer from spectral bias in capturing oscillatory and multiscale behaviors. In this study, we investigate this limitation by examining the failure of shallow ReLU neural networks in fitting high-frequency functions. This observation identifies two important factors in resolving rapid oscillations: the initial slope scale and the distribution of partition points induced by the networks. Motivated by this analysis, we propose RepNN, a reparameterized neural network model with activation ReLU or tanh designed for high-frequency and multiscale problems. The key idea is to reparameterize the weights and biases in the first hidden layer, which enables effective control of the initial slope scale and provides an appropriate distribution of the initial partition points. Furthermore, treating the reparameterized weights and biases as trainable parameters allows the DNN to achieve adaptive frequency scaling during training. In addition, we derive quantitative estimates for the output and slope magnitudes of the reparameterized DNN to guide the initialization of the proposed method. Numerical experiments, including multiscale one- and four-dimensional function approximations, forward and inverse PDE problems in combination with physics-informed neural networks (PINNs), and operator learning for an earthquake problem using real data, demonstrate that RepNN improves the predicted accuracy of vanilla DNNs in capturing highly oscillatory features with slightly additional computational cost. These results indicate that RepNN provides an effective and flexible approach for overcoming spectral bias and applying DNNs to multiscale problems.

One way to construct Difference Distance Magic Oriented Graphs (DDMOGs) is via a recently introduced technique called weighted sum. We explore the quality of said construction further by introducing the notion of an ornament. An ornament is an oriented graph that, when used in a weighted sum with an existing DDMOG, creates a new DDMOG. We provide results on the construction of a specific type of ornaments, called s-nodes ornaments. We conclude the paper with a list of open questions related to DDMOGs and ornaments.

We study dynamical orthogonal (DO) approximations of stochastic differential equations and investigate their long-time behaviour. The DO formulation represents the solution by a low-rank decomposition and leads to a coupled system consisting of an evolution equation on the Stiefel manifold and a reduced stochastic process. We establish the well-posedness of the strong DO system and derive quantitative error estimates between the original stochastic differential equation and its low-rank approximation in the Wasserstein distance. Our main contribution is the analysis of invariant probability measures for the DO dynamics. Under suitable dissipativity, Lipschitz continuity, and non-degeneracy assumptions on the coefficients, we prove the existence of an invariant probability measure for the strong DO system. The proof combines uniform moment estimates, a Krylov--Bogoliubov argument for an associated frozen system, and a Kakutani-Fan-Glicksberg fixed-point theorem to recover the self-consistent dynamics. We further show that the induced low-rank process admits an invariant probability measure and discuss the structure of invariant measures through several illustrative examples. These results provide a rigorous foundation for the use of dynamical low-rank approximations in the approximation of long-time statistical properties of stochastic dynamical systems.

Empirical risk minimization on massive datasets naturally exhibits a nested double finite-sum structure, where $N=nm$ total samples are logically or physically partitioned into $n$ blocks of size $m$ (e.g., in pooled data silos, out-of-core learning, or deliberate stratification). While variance-reduced methods achieve optimal oracle complexities for nonconvex objectives, they suffer from severe scaling bottlenecks in this centralized regime. Recursive estimators, such as PAGE, require periodic global full-gradient refreshes over all $nm$ samples, which are computationally expensive. Conversely, single-loop methods, such as SILVER, avoid such refreshes but require an impractical $\mathcal{O}(nm)$ memory footprint to store a control variate for every sample. In this paper, we propose SILAGE, a variance-reduced algorithm that addresses this trade-off. By actively exploiting the double-sum structure, SILAGE eliminates periodic global full-gradient refreshes over all $nm$ components (evaluating at most one local group gradient per iteration) while requiring only $\mathcal{O}(n)$ memory. Furthermore, we provide a tight convergence analysis that avoids pessimistic worst-case Lipschitz constants. Instead, SILAGE's complexity natively adapts to the underlying data geometry via nested functional similarities: across-group ($δ_1$) and within-group ($δ_2$) heterogeneity. Our results improve existing state-of-the-art bounds in several practically relevant regimes.

The saturation number $μ^*(G)$ of a graph $G$ is the minimum cardinality of a maximal matching, and $H(G)$ is its harmonic index. TxGraffiti conjectured in 2023 that $μ^*(G) \le H(G)$ for every nontrivial connected graph $G$, and Bıyıkoğlu refuted this by showing that the ratio $μ^*(G)/H(G)$ can be made arbitrarily large. Restricting to trees bounds the ratio sharply. Every nontrivial tree $T$ satisfies $μ^*(T) < \frac{3}{2} H(T)$, with the constant $3/2$ best possible. A complementary bound $H(G) < 4μ^*(G)$ holds for every graph with an edge, so on a nontrivial tree the saturation number is pinned to $\frac{1}{4} H(T) < μ^*(T) < \frac{3}{2} H(T)$, both constants best possible. The friendship graph $F_4$ is a smallest counterexample to the conjecture, on nine vertices, and the smallest tree counterexample is the subdivided star on eleven vertices. For each positive integer $m$ a family of graphs with $m$ hubs has ratio approaching $m+1$, while the conjecture holds whenever all vertices have equal degree. Both invariants arise in applications, the harmonic index as a molecular descriptor and the saturation number as a measure of adsorption inefficiency, and the bounds estimate the latter, which is NP-hard to compute, by the former, which is computable in linear time.

We investigate the propagation profile of positive solutions to \begin{equation*} u_t-du_{xx}=f(t,u) \mbox{ for } t>0,\ x\in(g(t),h(t)), \end{equation*} where $f(t,u)$ is monostable in $u$ and $T$-periodic in $t$, and the free boundaries $x=g(t), \ x=h(t)$ are determined by the Stefan condition $g'(t)=-μu_x(t, g(t)),\ h'(t)=-μu_x(t,h(t))$, coupled with $u(t, g(t))=u(t, h(t))=0$. For a special nonlinearity satisfying the strong KPP condition, the long-time behavior and asymptotic spreading speed of this problem were considered by Du, Guo and Peng \cite{DGP}. In this paper, by employing new techniques, we extend the results of \cite{DGP} to general monostable nonlinearities beyond the KPP framework and at the same time we obtain more precise description of the propagation profile: we prove the existence and uniqueness of a semi-wave and show that the spreading solution converges to this semi-wave as time goes to infinity.

We establish matrix versions of the comparisons between the $\ell^p$-norms or quasi-norms for sequences of complex numbers. For instance, given $1\ge q>0$, and a family of $m$ normal $d\times d$ matrices $A_1,\ldots, A_m$, we show that $$ \left|\sum_{k=1}^m A_k\right| \le \frac{1}{d}\sum_{i=1}^d V_i\left\{\sum_{k=1}^m |A_k|^{q}\right\}^{1/q}\!\!\!\!V_i^* $$ for some unitary $d\times d$ matrices $V_1,\ldots, V_d$. We also give applications to Olson's spectral order and to the comparison between the symmetric modulus and the quadratic symmetric modulus. In particular we show that the sum $A+B$ of two positive matrices submajorizes their Kato supremum $A\vee B$, thereby completing majorization results due to Ando.

We give an exact finite-dimensional Perron--Frobenius realization of the dominant zero of the Nekrasov--Okounkov polynomials $\nop _n(z)$. For a normalized positive sequence $h=(h(n))_{n\ge 1}$ with $h(1)=1$, define $\pol _0^h(z)=1$ and, for $n\ge 1$, \[ \pol _n^h(z)=\frac{z}{h(n)}\sum_{k=1}^n σ(k)\pol _{n-k}^h(z),\] where $σ(k)$ denotes the sum of divisors of $k$. The Nekrasov--Okounkov polynomials are obtained from the specialization $h(n)=n$ by the shift $\nop _n(z)=\pol _n^h(z+1)$. We derive a Hessenberg determinant representation for $\pol _n^h(z)$. After separating the trivial zero at the origin, the remaining zeros of $\pol _n^h(-z)$ are identified with the eigenvalues of an explicit $(n-1)\times(n-1)$ nonnegative matrix $M_n^h$. We prove that $M_n^h$ is primitive and apply Perron--Frobenius theory to show that $\pol _n^h(z)$ has a unique zero of maximal modulus; this zero is real, negative, and simple. As a consequence, the same property holds for the Nekrasov--Okounkov polynomials. We also prove strict monotonicity of the associated spectral radii.

Hamiltonian partial differential equations (PDEs) often exhibit long-time dynamics governed by conserved quantities such as mass, momentum, and Hamiltonian energy. Standard Fourier neural operators (FNOs) provide efficient data-driven approximations of solution operators, but may not preserve these invariants during autoregressive rollout, and can develop drift in conserved quantities, phase error, and loss of qualitative accuracy. We propose an energy-projection Fourier neural operator (EP-FNO), a structure-informed operator learning architecture that combines a residual FNO time-stepping update with an invariant projection for long-time prediction of parametric Hamiltonian PDEs. We also provide a theoretical analysis showing that EP-FNO can approximate operators associated with PDEs efficiently, we also suggest a stability estimate. We evaluate the approach on the Zakharov--Kuznetsov, Kadomtsev--Petviashvili, and sine--Gordon equations. Numerical experiments show that the projected model improves long-time stability, and gives more accurate propagation of soliton and coherent wave structures compared with a standard FNO baseline. Our results demonstrate that invariant projection improves the reliability of learned surrogates for long-time Hamiltonian PDE simulation.

Good and Meddaugh proved that every compact metric dynamical system with shadowing is a factor of the inverse limit of an inverse sequence of shifts of finite type. We show first that, for this factor representation alone, both assumptions are unnecessary: every compact Hausdorff dynamical system is a factor of the inverse limit of an inverse system of shifts of finite type. In particular, the mere existence of such a symbolic inverse-limit representation is not specific to shadowing. The main contribution of the paper is to identify the additional stability which shadowing provides in the metric case. We prove that every compact metric system with shadowing is a factor of the inverse limit of an inverse sequence of shifts of finite type whose bonding maps are surjective. Hence the inverse sequence satisfies the Mittag-Leffler condition, and the corresponding zero-dimensional extension still has shadowing. This strengthens the metric representation theorem of Good and Meddaugh and completes their characterization in terms of ALP factors of Mittag-Leffler inverse sequences of shifts of finite type. Finally, for arbitrary compact Hausdorff spaces, we show that every compact shadowing system is conjugate to the inverse limit of metrizable shadowing systems with factor bonding maps. In this sense, compact shadowing systems are generated from shifts of finite type by applying, at most three times, the two shadowing-preserving operations of taking Mittag-Leffler inverse limits and passing to ALP factors.

We prove that many virtual properties are closed under taking graph products, including: virtually RFRS, virtually (compact) special, virtually CAT(0) cube, and virtually normally poly-free. Our proof uses Januszkiewicz and Świątkowski's strong commensurability theorem for graph products, for which we provide an elementary proof.

We develop a control-theoretic framework for understanding Quantum Advantage (QA), providing a systematic route to characterize when and how QA can arise. The bilinear controlled Schrödinger equation is the common thread: the target quantum computation is recast as an operator controllability problem on the special unitary group $SU(N)$, and QA is identified with a polynomial-in-$n$ upper bound on the associated minimal-time function. We illustrate the framework on two paradigmatic problems: a) the Quantum Fourier Transform (QFT) on superconducting digital quantum processors (such as IBM's ibm_brisbane), for which we prove operator controllability by a Lie-algebraic argument and derive an $O(n^2)$ upper bound on the minimal time via a gate-concatenation lemma combined with the standard QFT circuit decomposition; b) the Maximum Independent Set (MIS) problem on neutral-atom analog quantum processors (such as Pasqal's hardware), for which we analyze the Rydberg-blockade Hamiltonian as a bilinear control system and reformulate the Quantum Approximate Optimization Algorithm (QAOA) as a continuous-time optimal control problem. By a controllability result, we show how the problem can be solved on Pasqal Quantum Computers and we introduce a control-based definition of Quantum Advantage for MIS. We conclude by outlining several open problems that chart directions for future research at the intersection of Control Theory and Quantum Computing.

We prove that two fixed sources in the Euclidean plane can be realized by a bounded-degree planar unit-edge graph on a 10-net, with graph distance from each source agreeing with Euclidean distance up to a universal additive constant. We ask whether the analogous statement holds for three non-collinear sources, and prove a logarithmic obstruction for large ordered source sets in the coordinate-planar setting.

We study the initial-value problem for the Schrödinger map flow from flat torus $\mathbb{T}^d$ into compact Kähler manifold $\mathcal{N}$. When $d \geq 3$ and $\mathcal{N} = \mathbb{S}^2$, we establish local well-posedness in $H^σ_x$ with $σ> d/2 + 1/2$. In this case, the evolution equation for the gradient of the solution reduces to a certain semilinear nonlinear Schrödinger equation (also known as modified Schrödinger map flow) when formulated in orthonormal frames. For general compact Kähler targets, we only obtain local well-posedness in $H^σ_x$ with $ σ> d/2 + 5/6$ due to the quasilinear nature of the flow, but in all dimensions $d \geq 2$. To the best of our knowledge, this is the first low-regularity local well-posedness result for Schrödinger map flow in the periodic setting, which yields a gain of $1/2$ derivatives for $\mathbb{S}^2$ targets and $1/6$ derivatives for general Kähler targets compared to the classical results \cite{DW,M}. The key ingredients of our method are an $L_{t, x}^2$ bilinear estimate for the first case and an \emph{a priori} $L_t^6L_x^{\infty}$ estimate for the second case, which are both achieved by combining the mass/energy and momentum balance laws of the equation with a new type of div-curl lemma introduced by the second author.

We extend the theory of root clusters from perfect fields to general fields which are not necessarily perfect. We introduce the following notions for field extensions over any given base field and study their interesting properties: root cluster size, multicluster size and their generalizations root capacity, multiroot capacity; ascending index, ascending normal index and their generalizations intersection indicium, intersection normal indicium; compositum indicium and compositum normal indicium. We establish our results on the Inverse problems for these generalized notions over Hilbertian fields which generalizes our earlier results which were over number fields. In particular, we show over a given Hilbertian field, the existence of a polynomial for given degree, cluster size and multicluster size and existence of an extension for given root capacity and multiroot capacity with respect to that polynomial.

We show that for any $A\subseteq \mathbb{F}_q^n$ lacking non-trivial solutions to a translation-invariant linear equation of genus one, meaning that no nonempty proper subset of the coefficients sums to $0$, there is a set $B\subseteq \mathbb{F}_q^m$ in some higher dimension which also lacks non-trivial solutions, such that \[|B|^{1/m}>|A|^{1/n}.\] In particular, this implies that no fixed cap set in $\mathbb{F}_3^n$ gives an asymptotically optimal lower bound by direct products alone.

For nearly a century, the transition state has been thought to lack an exact quantum counterpart: recrossing-free, one-way flux seems to require simultaneous knowledge of position and momentum. We show that this obstruction is illusory. The exact quantum flow contains a transition-state geometry: stable and unstable manifolds meeting in a unique bounded quantum transition-state trajectory that anchors a dividing surface carrying one-way quantum probability flux. The geometric framework of classical reaction dynamics survives in exact quantum mechanics, in a fundamentally quantum form.

Could two worlds localised on neighbouring branes communicate through gravity alone? We investigate this question in a minimal higher-dimensional framework in which Standard Model fields are confined to our brane while gravity propagates through the bulk. From the brane-to-brane graviton propagator we derive the retarded transfer kernel of the inter-brane link and identify the transition from evanescent to propagating Kaluza-Klein modes. The central idea is to give the Kaluza-Klein tower a new role: not only as a spectrum of massive gravitational states, but as a set of communication carriers. Below the first KK threshold the channel is effectively four-dimensional and is mediated only by the massless graviton. Above threshold, massive KK modes open as additional propagating subchannels, and information may be encoded in their occupation pattern, relative phases, and arrival-time structure as well as in ordinary signal variables. The compactification determines the KK masses, wavefunctions, brane overlap factors, and propagation phases, which together define a multi-input multi-output (MIMO) channel matrix. In the resolved-mode limit, the tower yields approximate parallel subchannels, to which standard information-theoretic notions such as capacity bounds, water-filling, effective rank, and sparse occupancy codes apply. The production and detection of such signals are highly model-dependent and not assumed to be feasible with known technology. Nevertheless, the channel structure is well defined: a neighbouring brane-world could be separated from us by a microscopic distance in the compact space while remaining hidden because the only shared interaction is gravity. The first observable signature may not be a deliberate message, but the spectral and modal structure of the Kaluza-Klein tower itself, revealing partial information about the geometry of a nearby hidden world.

We develop indexed Bellman information complexity, a representation-level theory of interactive decision making centered on information indices and reference histories. The representation strips away problem-specific syntax and retains only the ingredients needed for dynamic programming and information accounting, thereby unifying the earlier framework of indexed algorithmic information ratios (AIR). On the upper-bound side, regret is controlled by Bellman supersolutions or potential identities whose gradient bracket is paid for by indexed information. Upper-confidence-bound (UCB), estimation-to-decision/decision-estimation-coefficient (E2D/DEC), and adaptive-minimax-sampling or exploration-by-optimization (AMS/EBO) methods appear as three relaxations of this same identity. On the lower-bound side, the posterior-reference trajectory supplies both the information telescope and the ghost quantile of small-regret trajectories. The resulting critical radius in the lower bound is an effective-dimension-scale quantity, as in Fano and local-prior-mass lower bounds, rather than the constant radius of a two-point Le Cam argument. The examples show that DEC is best viewed as a one-step relaxation of indexed Bellman information complexity, not as a universally tight conversion mechanism. We illustrate the framework through several applications, with particular emphasis on kernel bandits. In this setting, the active action marginal provides a concrete basis for comparing UCB, E2D, and AMS/EBO.

Recently, Starr used asymptotic methods to disprove a conjecture by Heim--Neuhauser and Abdesselam about the log-concavity of the D'Arcais polynomials, without giving an explicit counterexample. We refine the asymptotics, to give the necessary estimates on convolutions of $σ_{-1}$, and identify the first counterexample at $λ= 65\,214\,507\,758\,400$. We also consider the asymptotic density of such counterexamples.

A finite group is said to be quartic-free if its order is not divisible by $p^4$ of any prime $p$. A finite group is called an $A$-group if all of its Sylow subgroups are abelian. Objective of this paper is to provide explicit structure of a quartic-free $A$-group. Further in the process of providing the explicit structure we also determine the derived length of a solvable quartic-free $A$-group.

Both tensor trains (TTs) and quantum states provide compressed representations of grid-structured data with potentially exponential compression power. We present a unified framework for upsampling data encoded in vector amplitudes, with efficient realizations in both classical TT and quantum settings. Starting from an \(n\)-core TT or an \(n\)-qubit state on a coarse grid with \(2^n\) points, the construction produces an \((n+m)\)-core TT or \((n+m)\)-qubit state on a finer grid with \(2^{n+m}\) points. In the TT setting, it supports interpolation, quasi-interpolation, augmentation, and synthesis through efficient low-rank contractions, with the added \(m\) cores retaining constant rank. For function-value encodings, the resulting interpolation satisfies an \(\ell^2\)-error bound independent of the number of added grid points, achieves exponential compression at fixed accuracy, and has a logarithmic complexity in the number of grid points. In the quantum setting, the refined state is prepared by a \(\mathrm{poly}(n,m)\)-size circuit using \(\log(p+1)\) ancillas, where \(p\) controls the smoothness of the quasi-interpolant; the corresponding error scales quadratically with the initial grid spacing. We validate our framework for tensor networks in one-, two-, and three-dimensional examples, including functions, derivatives, airfoil masks, and synthetic random fields such as three-dimensional turbulence. In particular, fractal fields can be generated directly in TT format with logarithmic memory and runtime. These results open a practical route to multiscale solvers, generative models, and geometry-aware algorithms on tensor-network and quantum platforms, with potential applications in scientific simulation, imaging, and real-time graphics.

This manuscript contains preprint of a chapter under consideration for inclusion in the forthcoming third edition of {\em Cover and Thomas's Elements of Information Theory}, posted with permission from Wiley. The table of contents EIT-3 ToC of the new edition can be found at: https://docs.google.com/document/d/1L-m4oQEJw1PJhoxBeMwrrBD8S_HmvzMEkPbYvS24980/edit?usp=sharing . For feedback, please contact abbas@ee.stanford.edu Learning and information theory intersect in both model training and the characterization of fundamental performance limits. This manuscript provides a concise and accessible treatment of the first intersection, requiring only basic background in information theory and statistics at the senior undergraduate or first-year graduate level. End-of-chapter exercises make the material well suited for classroom use as well as self-study. The chapter focuses on the role of divergence measures in model training, with examples ranging from linear and logistic regression to autoregressive models, variational autoencoders, diffusion models, generative adversarial networks, and score-based models. It introduces the evidence lower bound (ELBO), f-divergences, and the Fisher divergence. In particular, the treatment of the generative diffusion model provides a more systematic and explicit derivation than is typical in the literature.

We introduce the degree filtration on the discrete cubical chain complex of a graph, defined in terms of the maximal injective dimension of the facets of singular $n$-cubes, and study the degree spectral sequence which arises from this filtration. This spectral sequence interpolates between the discrete cubical homology of a graph $H_n(G)$ and the injective homology $H_n^{inj}(G)$, a variant of the discrete cubical homology based on injective singular cubes. Building on the work of Babson et al. we introduce the combinatorial condition of quasimonophobicity on graphs, and show quasimonophobicity implies both the vanishing of the degree spectral sequence in certain bidegrees, and implies $H_n^{inj}(G)$ is isomorphic to the homology of the CW complex obtained by ``filling in'' subcubes of the graph. These results are applied to compute $H_2(G_n^{sph})$ for the Greene sphere graphs $G^{sph}_n$.

We consider a general class of ``inexactly smooth'' convex functions, providing a universal model capturing as special cases $L$-smooth, $M$-Lipschitz, and Hölder smooth functions, and any combination thereof. Such functions possess a calculus closely following that of smooth functions. Our main results provide inexactly smooth functions with interpolation theorems that are necessary and sufficient up to modest universal constants. These enable analysis of first-order methods for any inexactly smooth convex problem class via solving convex Performance Estimation Problems (PEPs). Further, these enable the extension of Drori and Taylor's constructive approach to algorithm design. From this, we derive an exactly minimax optimal method for $(β,0)$-Hölder smooth problems, methods with the best-known convergence guarantees up to constants for any $(β,p)$-Hölder smooth convex minimization, and a new universal fast backtracking method for any inexactly smooth convex problem.