Papers

Characterizing fade duration in wireless channels is fundamental for designing robust communication systems. Classical approaches – Rice's level-crossing theory and Monte Carlo simulation – lack precision for tail events and are computationally prohibitive for rare-event probability estimation. This paper introduces a rigorous framework combining Wiener Chaos Expansion (WCE), Malliavin Calculus, and importance sampling with adaptive weights to analyze fade duration $Z(T)$ distributions. Main contributions include: (i) high-accuracy moment estimation and CCDF characterization via WCE minimizing Monte Carlo variance; (ii) Markovian projection reducing infinite-dimensional dynamics to tractable systems ($\dim \leq 3$) for Rayleigh, Rician, and Nakagami models under stated assumptions; (iii) asymptotically optimal importance sampling weights derived from Malliavin sensitivities, achieving 839 to 2516x variance reductions; (iv) a theoretically grounded and provably efficient adaptive SGD algorithm with Robbins-Monro step size schedule for parameter estimation. Numerical experiments validate our approach with relative errors below 0.5%, enabling gradient-based optimization of fade duration statistics even for regimes where $P \sim 10^{-15}$, without requiring $\mathcal{O}(1/P)$ samples, by evaluating sensitivities through analytical Malliavin weights.

We prove that if $f\in \mathbb Z[x]$ is a monic polynomial of degree $k\geq 2$, then there exists a constant $c>0$, depending only on $f$, and finite sets $A\subset \mathbb R$ of arbitrarily large size such that \[ |f(A)|\leq |A|^{k-c}, \] where $f(A)$ is interpreted in the Minkowski sum-product sense. In particular, taking $f(x)=x^2+x$, this gives a power-saving upper bound for $AA+A$, answering a question raised by Roche-Newton, Ruzsa, Shen, and Shkredov.

Symmetry provides a quantum neural network structure, but on its own it does not keep the network trainable once noise is present. We ask which physical quantity decides whether the gradients of an equivariant circuit survive decoherence, and we answer with a compact training law. Working with U(1)-equivariant brickwork circuits that conserve a charge, we find that two distinct effects govern a trainable gradient. Causality fixes where the gradient can live, confining it to the backward light cone of the readout inside the active charge sector. Coherence then determines how fast it decays through the contraction of the off-diagonal sector modes that the projected readout can actually observe. We prove a light-cone reduction that pins the noiseless gradient to the sector-restricted cone with a lower bound independent of the total qubit number, and we define a readout-visible aligned coherence rate as a Rayleigh quotient of the noise generator along the gradient-carrying mode. A perturbative open-system analysis turns this rate into a leading-order training law. Density-matrix simulations then confirm that the finite-noise degradation follows a single accumulated variable built from noise depth and coherence contraction, with a coefficient of determination of 0.979. The sharpest test comes from a correlated-dephasing channel that has a large worst-case rate but a near-zero aligned rate. The law predicts no gradient loss for this channel, and none is seen. Sector coherence outperforms every standard channel diagnostic we compare it against, and the analysis identifies readout-visible sector coherence as the quantity that links equivariant architecture, open-system dynamics and noisy trainability.

We produce a formula, analogous to the Gauss-Codazzi equation, which relates the geometry of a $G_2$-structure and its Hodge Laplacian to the geometry of the induced $SU(3)$-structure on an embedded hypersurface. As an application, we obtain necessary conditions for the solvability of the Poisson equation for (not necessarily closed) $G_2$-structures in a neighbourhood of this hypersurface. Next, we prove that our conditions are sufficient in the cohomogeneity one setting, assuming the symmetry group is compact and simple.

In this expository article, we present on state-of-the art results regarding three closely related invariants of moduli spaces of curves: their Chow rings, cohomology rings, and point counts over finite fields. We study the moduli space $\mathcal{M}_{g,n}$, parameterizing smooth genus $g$ curves with $n$ marked points, as well as its compactification by stable curves $\overline{\mathcal{M}}_{g,n}$. After explaining the relationship between these different invariants, we survey what is know regarding the following related questions: When are the Chow rings tautological? When are the cohomology groups tautological? And when are the point counts over fields of size $q$ given by a polynomial in $q$?

This paper investigates the persistence of maximum likelihood paths in degenerate stochastic differential systems and quantifies how small periodic perturbations modulate the metastable transition rate. Within the Freidlin–Wentzell large deviation framework, we reformulate the variational problem for MLPs as a Hamiltonian system via a partial Legendre transform. Under hyperbolicity and transversality conditions, we prove, using a geometric Melnikov method adapted to general time-dependent perturbations, that the corresponding heteroclinic connections persist for sufficiently small perturbations. For the periodic case, we derive a closed-form explicit expression for the rate change to first order in the forcing amplitude. Two illustrative examples are presented.

Large-scale combinatorial optimization is a challenge for near-term quantum computing because dense Quadratic Unconstrained Binary Optimization (QUBO) formulations yield interaction graphs that exceed the limits of NISQ processors. This work introduces Hybrid Quantum Neighborhood Selection (HQNS), a hybrid framework mitigating this via stochastic frontier decomposition. Instead of encoding all N variables into a monolithic circuit, HQNS selects a compact frontier of F << N active variables per stage, freezing the rest into reduced QUBO coefficients. A multi-stage crawling procedure rotates these frontiers, letting local quantum subproblems refine a global solution. We evaluate HQNS on the Maximum Diversity Subset Selection Problem (MDSSP) across six scales, N up to 1000. Circuit burden is reduced from the dense QAOA requirement of O(N^2) two-qubit terms per layer to O(F^2) per stage, with total complexity governed by the number of stages and classical overhead. Benchmarks show that HQNS achieves competitive solution quality relative to parallel simulated annealing (SA) while maintaining bounded circuit width and stable QPU time. In the N=1000 benchmark over ten executions, HQNS preserves 99.9908% of the mean diversity score of an 11-restart parallel SA baseline, while reducing wall-clock time by 65.03%, peak CPU usage by 55.97%, and peak memory by 35.21%. Ablation shows performance depends on frontier size, warm-starts, CVaR filtering, and stochastic rotation. These results demonstrate that structured frontier decomposition makes variational optimization executable for dense QUBO instances unsuitable for direct QAOA on present hardware.

The solutions of algebraic differential equations in certain valued differential fields, including the differential field of transseries, can be analyzed using a Newton diagram method. In this paper, we show that (eventual) equalizers, a crucial part of this process, can be obtained uniformly and definably from the coefficients of the input differential polynomials. We also obtain similar definability results for a certain compositional conjugation which is used repeatedly as an intermediate simplification step.

We classify all skew left braces with additive group isomorphic to the infinite dihedral group. There are ten isomorphism classes.

We study the siblings version of the coupon collector problem. A main collector stops when every coupon type has appeared at least once, duplicates are passed successively to later siblings, and $U_j^N$ denotes the number of empty spaces in collector $j$'s album at the main completion time. We prove three results. First, for every fixed $N$ and $j\ge2$, $\E U_j^N$ is uniquely maximized over positive coupon distributions by the uniform distribution; in fact it decreases strictly along every nonconstant ray from the uniform vector. Second, in the uniform model, $U_j^N$ is stochastically increasing in $N$, and we construct an increasing coupling using top spacings of exponential order statistics. Third, for fixed album indices $2,\ldots,J$, the naturally normalized vector converges jointly to $(W,\ldots,W)$, where $W$ is exponential with mean one. We also derive exact Poissonized and alternating-subset formulae and give a transfer principle for leading expectation asymptotics.

We establish an infinite-dimensional affine transform theory for the time-augmented Brownian signature. Our first main result shows that, for a suitable class of linear functions of the signature, the conditional Fourier-Laplace transform admits an entire signature expansion. We prove that the associated coefficients solve an infinite-dimensional linear differential equation on the extended tensor algebra. Our second main result shows that the logarithm admits a local signature expansion whose coefficients satisfy a Riccati equation on the extended tensor algebra, revealing a generalized affine structure of the Brownian signature in a genuinely path-dependent setting. In contrast to conventional affine processes, we show that this representation is intrinsically local: zeros of the Fourier-Laplace transform in the complex plane prevent any global expansion. To recover global representations, we introduce a new class of randomized Riccati equations with path-dependent terminal conditions through a recentering argument. Furthermore, we establish uniqueness of solutions to the linear and Riccati equations within a suitable class of solutions. Our results provide a theoretical framework for transform methods in non-Markovian settings, with applications to the computation of conditional distributions.

We design and analyse a family of hypocoercivity-preserving fully discrete Galerkin methods for the kinetic Fokker–Planck (kFP) equations, a class of evolution partial differential equations (PDE) with degenerate diffusion. The key idea in the method design is to mimic the framework of enhanced quadratic forms of Villani \cite{villani} to obtain a coercive bilinear form in a sufficiently strong exponentially weighted norm admitting a spectral gap/Poincaré inequality, despite the degeneracy. The problem is posed as a fourth-order evolution PDE on the whole space $\mathbb{R}^{d}\times\mathbb{R}^d$. The spatial discretization employs continuous piecewise polynomial approximation spaces on simplicial and/or box-type meshes comprising finite and infinite elements. Nonconformity is treated by constructing appropriate numerical fluxes in the spirit of $C^0$-interior penalty ($C^0$-IP) methods for fourth-order elliptic problems. The proof of coercivity requires new trace inverse inequalities for exponentially weighted norms on $d$-dimensional simplicial and box-type domains, and prismatic domains with infinite length. We prove these inequalities for a variety of exponential weights. Once coercivity is established, exponential convergence to equilibrium follows from an exponentially weighted Poincaré inequality. A fully discrete extension is obtained by coupling the spatial discretization with an $hp$-version discontinuous Galerkin time-stepping scheme. Exponential convergence to equilibrium is also proven for the fully discrete scheme. The proposed methods preserve total mass and exhibit provably exponential convergence to equilibrium, a manifestation of `numerical hypocoercivity,' making them well suited for long-time kFP simulations. Numerical experiments validate the theoretical results and investigate the convergence behaviour of the proposed method.

We study qualitative and quantitative homogenization for a Cahn–Hilliard system with a nonconservative source term in a periodically perforated domain. Using the periodic unfolding method, we derive uniform energy estimates and prove convergence to a homogenized Cahn–Hilliard system whose effective diffusion tensor is characterized by scalar Neumann cell problems on the pore cell. For the quantitative analysis, we construct first-order corrector approximations by means of a scale-splitting operator, so that the cell correctors are only required to belong to $H^1_{\mathrm{per}}(Y_p)$. Under $H^2$-regularity of the homogenized solution and well-prepared initial data, we obtain an order $\varepsilon^{1/2}$ corrector estimate: the corrected order-parameter error is controlled in $L^2(0,T;H^1(\Omega_p^\varepsilon))$, while the uncorrected order parameter is controlled in $L^2(0,T;L^2(\Omega_p^\varepsilon))$. This improves the rate $\varepsilon^{1/4}$ previously established for fourth-order phase-field equations in perforated media, and matches the natural rate for second-order elliptic problems in perforated domains. The rate reflects the boundary layer caused by incomplete cells near $\partial\Omega$ and improves to order $\varepsilon$ on the flat torus $\mathbb{T}^d$.

Cumulants are classical statistics associated with a random variable, defined as polynomial functions of its moments and distinguished by their additivity under convolution of distributions. Statistic is the name given to a function of a random variable, and a moment-based statistic is one that depends only on the moments $(\E[X^n])_{n\in \N}$. We prove a converse: any statistic depending continuously on finitely many moments and additive for independent sums must be a linear combination of cumulants. The proof uses an algebraic reformulation of the problem via the Hurwitz product and a linearizing change of coordinates. This result also follows from the more general theorem of Mattner \cite{mattner}, but our approach is elementary and self-contained.

ETSI Decentralized Congestion Control (DCC) limits roadside unit (RSU) broadcast rates based on channel load, yet its impact on age of information (AoI) for vehicle-to-infrastructure updates remains uncharacterized under real traffic. We derive the AoI of DCC-constrained V2I broadcast, revealing a hyperbolic density dependence that induces diurnal AoI variation exceeding 4 times on a four-RSU corridor, with the DCC target CBR parameter as the dominant control. We propose a cooperative policy exploiting upstream spatial traffic correlation to improve channel load estimation, with a safeguard ensuring non-negative gains. Evaluated on a 5-day, 762,050-vehicle trace from Kuwait City, the policy reduces corridor AoI by 5% at moderate and up to 66% at conservative DCC settings.

We prove a global \(C^{1+\text{H\"older}}\)-rigidity theorem for Anosov diffeomorphisms of tori with irreducible linearization. Let \(f:\mathbb T^N\to\mathbb T^N\) be a \(C^2\) Anosov diffeomorphism with linearization \(A\in GL(N,\mathbb Z)\), and assume that \(A\) is irreducible. If, for every periodic point \(p=f^n p\), the linear maps \(Df_p^n\) and \(A^n\) are conjugate, then the Franks–Manning conjugacy between \(f\) and \(A\) is \(C^{1+\text{H\"older}}\). Thus, in the irreducible case, periodic data completely characterize global \(C^{1+\text{H\"older}}\)-rigidity. The proof does not assume conformality, uniform quasiconformality, simplicity of the spectrum, or any restriction on Lyapunov multiplicities. The main ingredient is a new partial-to-global rigidity mechanism combining geometric and analytic arguments. We first obtain partial cocycle rigidity on canonical conformal layers inside the Lyapunov blocks by geometric methods, and then promote this partial rigidity to full regularity of the conjugacy along the Lyapunov blocks by analytic methods. The same method yields a local rigidity theorem for \(C^1\)-small \(C^{1+\text{H\"older}}\) perturbations of \(A\).

Mossel and Peres (2005) established a comprehensive framework for designing Bernoulli factories. Notably, they demonstrated that a single-variable function admits a finite-automata Bernoulli factory if and only if it is a rational function. Their Theorem 2.9 claims an extension of this result to multivariable functions, but it contains a subtle technical oversight in the application of Pólya's Theorem. We provide a direct counterexample: a rational function in three variables that admits a general Bernoulli factory but cannot be implemented by a finite-automata Bernoulli factory.

In 1937, Stefan Kaczmarz proposed a simple algorithm for solving systems of linear equations. This algorithm turned out to be the earliest known example of stochastic gradient descent, a ubiquitous computing paradigm that drives the training of modern AI models such as ChatGPT and Gemini. Now, those AI models have joined forces to discover the worst-case complexity of the Kaczmarz algorithm. This paper tells the story of how it happened.

Given a local gauge theory on spacetime with boundary, it naturally defines another gauge theory which can be regarded as a theory of the boundary values. For Lagrangian theories, it comes equipped with the presymplectic structure which can be used to define one or another version of Hamiltonian-like formulation of the initial model. This relation is especially manifest for AKSZ sigma models and more-generally gauge PDEs with compatible presymplectic structure in which case the boundary system is again a gauge PDE with presymplectic structure. In the context of (flat space) holography one is interested in boundaries at infinity, also known as asymptotic boundaries. The gauge PDE framework naturally extends to this setup, resulting in the notion of gauge PDE with asymptotic boundaries. Although this works perfectly well at the level of equations of motion, the extension to Lagrangian systems appears quite subtle because the presymplectic structure capturing the Lagrangian is divergent at the boundary. We show that any $Q$-cocycle in the bulk (and presymplectic structure in particular) determines a pair of compatible $Q$-cocycles of the boundary gauge PDE: the renormalized one of the same ghost-degree, and the anomaly cocycle of degree one lower. For the latter, the construction is somewhat analogous to the residue map known in the context of b-geometry. The general formalism is exemplified by scalar and Maxwell fields on AdS and Minkowski spaces. It turns out that in the AdS case the natural action determined by the anomaly presymplectic structure is precisely the one known as the holographic Weyl anomaly in the AdS/CFT context while its null-infinity counterpart was known in a few very particular cases only.

We construct the quantum geometric Langlands functor in the Betti setting via Whittaker coefficients. We show that the functor is compatible with the 2-Fourier-Mukai equivalence between sheaves of categories over 2-stacks $\operatorname{Ge}_{Z_G}$ and $\operatorname{Ge}_{\pi_1(\check{G})}$, which classify gerbes on $X$ with respect to the center $Z_G$ of $G$ and algebraic fundamental group $\pi_1(\check{G})$ of $\check{G}$.

Neural compression is currently dominated by Nonlinear Transform Coding (NTC), which maps data to real-valued latents via continuous transforms. Despite its success, NTC suffers from train-test mismatch due to non-differentiable quantization, a “smoothness bias" inherent in continuous transforms that precludes optimality for certain sources, and a loss of “shaping gain" due to the complexity of including high-dimensional vector quantization. We propose SoftBinary Coding (SBC), an end-to-end learning paradigm that bypasses these limitations by using a stochastic binary latent space. In the spirit of vector quantization, SBC employs discrete representations and compresses them through a novel fast binary channel simulation scheme, for which we provide a proof of rate optimality. Experimental gains on information-theoretic sources provide both theoretical and practical closure to NTC's limitations, establishing discrete binary structures as a viable path toward reaching optimal rate–distortion bounds. Surprisingly, SBC also achieves state-of-the-art performance on vector quantization of i.i.d. sources, exceeding Trellis Coded Quantization of the Gaussian source.

We study Turn-Based Deterministic Forward Games (TBDFGs), the subclass of turn-based deterministic zero-sum games in which no directed cycle contains actions controlled by both players. This forward condition is strictly weaker than acyclicity: recurrent behavior may be arbitrarily rich within one player's states, while mixed-player feedback cycles are excluded. Our main contribution separates two algorithmic consequences of this structure. First, we analyze the simple strategy-iteration method of [11,14], a generic method for TBSGs whose execution neither tests for nor uses the TBDFG property. We prove that this structure-oblivious algorithm nevertheless has a strongly polynomial guarantee on every TBDFG. In particular, it terminates after at most $O(n^6m^4\log^4 n)$ simplex pivot steps. Thus, the forward property acts as a structural certificate for convergence even when the algorithm is not informed that the input has this property. Second, when the TBDFG structure is known in advance, a backward SCC propagation algorithm is proposed that solves a sequence of deterministic-MDP subproblems and improves the bound to $O(n^3m^2\log^2 n)$ simplex pivot steps. Together, these results show that forward structure both regularizes the convergence of a general strategy-iteration method and supports a sharper structure-aware algorithm.

We prove that, for almost all even integers $N>0$, the set of Goldbach primes $\mathbb{P} \cap (N-\mathbb{P})$ has a level of distribution $1/6$. As applications, we show that almost all even integers $N>0$ can be written as the sum of two primes $p_1, p_2$ such that $p_1-p_2+1 \in \mathbb{P}_4$. We also prove an analogous result with $2p_1 p_2+1 \in \mathbb{P}_{13}$ for almost all integers $N>0$ with $6\mid N$.

The paper studies the graph algebra whose monomial basis is naturally indexed by simple graphs on a fixed set of vertices. This algebra is at the same time the algebra of pseudo-Boolean functions on the Boolean cube and a natural object of algebraic combinatorics, related to the Boolean lattice of subsets of the edge set of the complete graph. The main aim of the paper is to study two compatible representation-theoretic structures on this algebra: the action of the Lie algebra $\mathfrak{sl}_2$, arising from the operators of adding and deleting one edge, and the action of the pair group $S_n^{(2)}$, induced by the renumbering of vertices. It is proved that the graph algebra with this $\mathfrak{sl}_2$-action is isomorphic to a tensor power of the standard two-dimensional $\mathfrak{sl}_2$-module, and on this basis its decomposition into irreducible $\mathfrak{sl}_2$-modules is obtained. Primitive spaces, that is, the kernels of the edge-deletion operator on rank components, are also described, and it is shown that they have a natural interpretation in terms of two-row Specht modules. It is then established that the $\mathfrak{sl}_2$-action commutes with the action of the pair group. It follows that the space of graph invariants also inherits the structure of an $\mathfrak{sl}_2$-module. Using Schur–Weyl duality, primitive invariants are described through the fixed parts of the restrictions of two-row Specht modules from the full symmetric group on the edge set to the pair group. As a consequence, the classical enumeration of non-isomorphic graphs by the number of edges receives a representation-theoretic refinement: the orbital components entering the Burnside–Polya formula decompose into natural primitive contributions associated with the $\mathfrak{sl}_2$-structure and two-row Specht modules.

We use probabilistic and stochastic-analysis methods to prove trace-norm propagation of chaos for finite-dimensional quantum mean-field systems governed by Belavkin equations. The particles are density matrices, interact through a mean-field Hamiltonian, and are continuously monitored through independent diffusive observation channels. The limiting dynamics is a nonlinear matrix-valued McKean-Vlasov diffusion, random through its local observation record and coupled through the deterministic averaged state. The main result treats arbitrary one-particle density matrices, including mixed states, and both perfect and inefficient measurement regimes. Under strong tensorization of the initial data, every fixed marginal converges uniformly on compact time intervals to the tensor product of the nonlinear limiting filters, with an explicit quantitative bound. The proof combines purification, fully observed dilation, conditional expectation, relative entropy, and uniform stability of the associated Zakai equations. In the skew-adjoint measurement case, exterior observation noises disappear from the marginal equations and a stochastic BBGKY hierarchy is recovered. Under only marginal chaoticity of permutation-invariant initial states, we prove convergence of fixed marginals by an iteration of this hierarchy.

We record a Weyl-positive reduction and certificate framework for the Riemann phase kernel associated with the even Riemann kernel $\Phi$. The manuscript does not present a complete proof of the Riemann hypothesis. Its immediate analytic target is a concrete positivity theorem for a Weyl kernel whose quantum characteristic function satisfies the Kastler–Loupias–Miracle-Sole condition in all numerical tests performed so far. Several natural factorizations are ruled out. In particular, the positive anti-Wick density route is obstructed by a local heat-deconvolution test, and several natural finite-core reductions are excluded by explicit counterexamples. The surviving structure is a finite-core Volterra program upgraded to a closed-trace quotient certificate for the full kernel. We derive exact same-sign finite-core formulae, the second-order theta-mode identity $\phi_n(t)=(\partial_t^2-1/4)(e^{t/2}e^{-\pi n^2e^{2t}})$ for $n\ge1$, a Volterra boundary-plus-tail representation, and a quotient Schur factorization for the normalized full-$\Phi$ source/Volterra model. The latest certificate closes the active trace-range condition, the full-continuum source-inactive domination, and the Douglas/Moore–Penrose Schur hypotheses in the normalized model. What remains outside that certificate is explicitly separated: the quotient-to-original Weyl lift, uniform $\omega$-coverage for $|\omega|<1/2$, and the final bridge from Weyl/KLM positivity to the intended de Branges or RH-side formulation.

Shuffle order can be a larger source of fine-tuning noise than a memoryless analysis predicts: fixed-clock optimizer memory makes local equal-multiset contrasts first order in the learning rate rather than second order, and the resulting order channel can be large enough for a single seed to flip a close A/B comparison. We isolate this mechanism and derive a fit-free way to size the noise it produces. For a memoryless optimizer, reordering an equal multiset has no first-order endpoint term; the leading local contrast is the $O(\eta^2)$ gradient bracket. Fixed-clock optimizers such as AdamW are different. Their moment buffers, preconditioner state, and de-biasing counters advance with the step index rather than with the learning-rate-scaled time $\tau=\eta k$, so the same gradient can receive a position-dependent endpoint weight. For any fixed finite measurement window, a lifted-state expansion gives an $O(\eta)$ equal-multiset contrast whenever the first-order replay coefficient is nonzero, while regular and clock-matched controls remain $O(\eta^2)$; a bare fixed-$\beta$ momentum buffer is already enough. A bitwise-deterministic replay from one warmed optimizer state isolates the mechanism, giving order-variance slopes 1.83 for AdamW, 2.00 for fixed-$\beta$ momentum, and 4.00 for SGD; matching the memory clock to $\tau$ restores the regular exponent. For AdamW with a frozen preconditioner, the same impulse-weight kernel gives a closed-form asymptotic order-variance floor after the local potentials are measured, with no fitted coefficients. The result is local to the measurement window (independent reshuffling can average the channel across windows), but it yields order-noise error bars, positional attribution weights, and a seed-budget criterion for fine-tuning comparisons.

The independence number $\alpha$ of a finite simple graph lies between the polynomial-time computable Havel-Hakimi residue $R$ and annihilation number $a$. The Caro-Wei sum $W$ gives another lower bound and the known upper bound $\alpha\le(\Delta+1)W/2$ for maximum degree $\Delta\ge1$. This upper bound does not by itself control the larger quantity $a$. This note proves $a\le(\Delta+1)W/2$ for $\Delta\ge1$, with equality attained, for example, by regular graphs of positive degree and even order. Together with $R\ge W$, this bound shows that the interval from $R$ to $a$ improves or matches both Caro-Wei endpoints. Its sharp worst-case endpoint ratio remains $(\Delta+1)/2$, even among connected graphs. For maximum degree at least three, the bound also proves the inequality $\alpha\ge(a+R)/\Delta$ conjectured by TxGraffiti, an automated conjecturing program. A direct calculation handles connected graphs of maximum degree two. Together these cases resolve the conjecture for every connected graph with at least three vertices. The single edge is the only nontrivial connected exception. Equality holds for exactly eight connected graphs up to isomorphism. For maximum degree at least three, the conjectured lower bound never exceeds the residue, so it gives no stronger independence estimate.

We prove that every entire smooth strictly plurisubharmonic solution of the self-shrinking $J$-equation on $\mathbb{C}^n$ is a quadratic polynomial. This removes the asymptotic lower bound assumption on the complex Hessian in \cite[Theorem 4]{HJ}. The result also recovers the corresponding real rigidity theorem in \cite[Theorem 1.1]{HOW} as a special case. More generally, our method applies to self-shrinking equations associated with a broad class of fully nonlinear elliptic operators satisfying suitable structural conditions, including the adjacent inverse complex Hessian quotient operators $-\sigma_{k-1}/\sigma_{k}$, $1\leq k\leq n$. For the remaining quotients $-\sigma_l/\sigma_k$ with $l<k-1$, we prove the same rigidity conclusion under the additional assumption that the solution is proper.

Given a 3-term perfect complex E over a quasi-projective variety X and a nonnegative integer r, we define two virtual cycles and their refinements supported over the r-th degeneracy loci of E. This is done by modifying the complex E after pulling it back to certain blow ups of X. We establish several Thom-Porteous, comparison, duality and wall-crossing formulas for these virtual cycles. We apply this construction to perfect complexes arising from the universal objects over the Picard variety and the Hilbert schemes of non-singular complex projective surfaces. We recover, reprove and strengthen some of the known results involving the reduced cycles and the virtual cycles of the Hilbert schemes related to the curve counting theory and Vafa-Witten theory, respectively. In the case of elliptic surfaces, we provide an explicit calculation generalizing that of Seiberg-Witten invariants.