arXiv subjects

Nikita Kalinin

Publications and source records attributed to Nikita Kalinin.

At least 19 recordsLinked to original sources

arXiv2026

Shrinking dynamic on multidimensional tropical series

Let $Ω\subset\mathbb R^n$ be a compact convex domain. An $Ω$-tropical series is a nonnegative, concave, integral-slope, piecewise-affine function on $Ω$ that vanishes on $\partialΩ$. For a finite set $P\subsetΩ^\circ$, we study the least such function above prescribed initial data whose corner locus contains $P$. It is obtained by repeatedly applying one-point shrinking operators $G_p$. We prove that every fair order of these operators stabilizes after finitely many nontrivial steps. We also describe an event-driven implementation that records the lowest monomials at each point and updates only affected watcher lists. Finally, we show that, on every compact subset of $Ω^\circ$, the resulting dynamics can be approximated by a finite path whose intermediate tropical hypersurfaces have only mild singularities on that compact set; equivalently, the corresponding local cells of the dual regular subdivision contain no lattice points other than their vertices.

math.AG↗

arXiv2026

The 2-Adic Valuation of the Order of the All-Ones Class in the Sandpile Group of a Square

Place one grain at every nonsink vertex of the wired $n\times n$ square, and let $L(n)$ be the order of this operation in the sandpile group. Thus $L(n)$ is the least positive $q$ for which $q$ uniform grain layers form an integral combination of toppling moves. We prove that, for every $n\ge1$, \[ ν_2(L(n))= \begin{cases} 2,&n=1,\\ 1,&n\ge2\text{ even},\\ ν_2(n+1)+2,&n\ge3\text{ odd}. \end{cases} \] For even squares, this follows from the domino--sandpile results of Florescu, Morar, Perkinson, Salter, and Xu, completed by a short parity observation. For odd squares, a unimodular cyclic basis identifies the folded cokernel with a quotient by two shifted Chebyshev polynomials and sends the all-ones class to $1$. Its order is determined by the constant part of this polynomial ideal, not just by a determinant. Two normalized Euclidean remainders reduce to consecutive Fibonacci polynomials over $\mathbb F_2$, giving the exact valuation.

math.CO↗

arXiv2026

Many-point tropical relaxation and the Monge--Ampère equation

We prove a quantitative tropical approximation to the planar Aleksandrov Monge--Ampère equation. Let $Ω\subset\mathbb R^2$ be a bounded open convex domain, fix $K\SubsetΩ$, and let $F_N=G_{P_N}(0_Ω)$ be the minimal nonnegative concave tropical series with integral slopes, zero boundary values, and corner locus containing a universally generic $N$-point set $P_N\subset K$. Set $u_N=N^{-1/2}F_N$ and $μ_N=N^{-1}\sum_{p\in P_N}δ_p$. For every compact $L\SubsetΩ$ we prove $\left|\intφ\,d(\mathrm{MA}(u_N)-μ_N)\right|\le C(Ω,K,L)N^{-1/2}(\|φ\|_\infty+\|\nablaφ\|_\infty)$ for $φ\in C_c^1(Ω)$ supported in $L$. If $μ_N\rightharpoonupμ$, where $μ$ is a probability measure supported in $K$, then $u_N$ converges uniformly on $\overlineΩ$ to the unique continuous concave zero-boundary Aleksandrov solution of $\mathrm{MA}(F)=μ$, and $\mathrm{MA}(u_N)\rightharpoonupμ$ vaguely in $Ω$. No regularity or strict convexity of $\partialΩ$ is assumed. For bounded rational convex polygons, strong genericity suffices. If $P\subset K$ is strongly generic with $|P|=N$ and $F_P=G_P(0_Ω)$, its tropical curve has exactly $N$ bounded cells; the duals of the uncut marked carriers form a spanning tree; every compact internal edge has weight one; and $\mathrm{MA}(F_P)(Ω^\circ)=N-1+\tfrac12D_{\mathrm{term}}(F_P)$, with $D_{\mathrm{term}}(F_P)=O_{Ω,K}(\sqrt N)$. For strongly generic sequences satisfying the same empirical-measure hypothesis, the normalized curvature measures converge weakly on the closed polygon. We also obtain almost-sure limits for i.i.d. samples from absolutely continuous laws supported in $K$, affine covariance of the continuum solution, and, for source sequences covered by the polygonal theorem, a configuration-dependent Abelian-sandpile diagonal.

math.AP↗

arXiv2026

Limits of Equi-Affine Equi-Distant Loci of Planar Convex Domains with Two Non-Parallel Asymptotes

In this note, we introduce equi-affine invariants by averaging over the space of tropical structures of fixed covolume. Applied to the tropical distance series, this construction produces a family of equi-affine invariant functions associated with convex domains which are expected to satisfy a number of remarkable properties. We conjecture a limiting description of the associated level sets in the compact case, and we prove an analogue of this statement for unbounded domains with two non-parallel asymptotes, showing the universality of the affine curvature of the resulting limiting hyperbola. In addition, we give an explicit formula for the arithmetic mean value at the center of the unit disk.

math-ph↗

arXiv2026

Topology of a Uniform Spanning Tree on a Cylinder

We study uniform spanning trees (USTs) on the discrete cylinder $G_{n,m}=C_n\times P_m$, in the regime where the circumference $n$ is fixed and the length $m$ tends to infinity. Using Wilson's algorithm, with the initial root on one boundary ring and the first walk started from the other, we single out a \emph{trunk} $L\subset T$: the first loop-erased random-walk path connecting the two boundary rings. We prove that the tree-distance from any fixed vertex to this trunk has an exponential tail with constants independent of $m$. As a consequence, the longest branch attached to the trunk is at most logarithmic in the length of the cylinder with high probability. Our motivation comes from the Abelian sandpile model on cylinders, and in particular from the step-like, or ``ladder'', avalanche-size distributions observed numerically by Eckmann--Nagnibeda--Perriard in \cite{EckmannNagnibedaPerriard2023}. Via Dhar's burning algorithm, recurrent sandpile configurations correspond to spanning trees, suggesting that the geometry of a typical UST may influence how avalanches propagate along the cylinder. The branch estimates above, their wired analogue, and the exponential estimate for the interface separating vertices whose paths to the sink pass through opposite boundary rings are intended as a first step towards isolating geometric UST observables that may be relevant to these plateau phenomena.

math.CO↗

arXiv2026

Weighted error-sum identities for periodic continued fractions and their generalizations

For a purely $N$-periodic continued fraction $ξ=[\overline{a_0,a_1,\dots,a_{N-1}}]=[a_0,a_1,\cdots]$, with $a_k=a_{k+N}$ for all $k\ge 0$, and convergents $h_n/k_n=[a_0,a_1,\dots,a_n]$, we obtain explicit expressions for the weighted error sums $f_ξ(s)=\sum a_{n+1}\lvert h_n-ξk_n\rvert^s$ for $s>1$. A key observation is that, for each residue class $k_0\in{0,1,\dots,N-1}$, the subsequence of approximation errors $(h_k-ξk_k)$ with $k\equiv k_0 \pmod N$ forms a geometric progression. In addition, we extend our methods to generalized continued fractions with numerators $(b_n)$, obtaining Euler-type identities and weighted error-sum formulae for $π$ and $\ln 2$.

math.NT↗

arXiv2026

A $p$-adic ($p\equiv 3\pmod 4$) depth-$5$ supercongruence for Gaussian $p$-th power sums over a square

For an odd prime $p$, define $G_n(p)= \sum_{a=1}^{p-1}\sum_{b=1}^{p-1}(a+bi)^n \in \mathbb Z[i]$. We study the $p$-adic valuation of these Gaussian power sums for $n\leq p, n=r(p-1)$ and show that it is governed by the interaction of the fourfold symmetry of the square, ordinary power-sum congruences, and Bernoulli numbers. If $p\equiv3\pmod4$ and $p\ge7$, then we prove an unexpectedly deep supercongruence $G_p(p)\equiv -\frac{p^5}{12}(p-1)^2(p-2)(1-i)B_{p-3} \pmod{p^6}$.

math.GM↗

arXiv2026

Proof of Sun's conjectures on hyperbolic cosine series via the Eisenstein--Lambert method

We prove two conjectures of Zhi-Wei Sun concerning hyperbolic cosine Lambert series. The first one is the evaluation, for integers $m\geq 0$, of \[ S_m=\sum_{n=1}^\infty\left( \frac{n^{2m}}{\cosh(πn)-1} -\frac{(2^{2m+1}-(-1)^{m(m+1)/2}2^{m+1}+4)n^{2m}}{\cosh(2πn)-1} +\frac{2^{2m+2}n^{2m}}{\cosh(4πn)-1} \right). \] We prove that \[ S_0=\frac1{12},\qquad S_1=\frac1{2π^2},\qquad S_m=0\quad (m>1). \] The second one is the quadratic identity \[ \sum_{n=1}^\infty \left( \frac{4}{(\cosh(πn)-1)^2} -\frac{55}{(\cosh(2πn)-1)^2} +\frac{16}{(\cosh(4πn)-1)^2} \right) = \frac{77-234/π}{72}. \] The proof uses an elementary level-four identity for Lambert series and its consequences for Eisenstein series. After differentiating this identity and evaluating it at $i/2$, the first conjecture follows from the modular transformation law for $E_{2m}$, with the cases $m=0$ and $m=1$ treated separately by the quasimodular transformation law for $E_2$. The second conjecture follows by rewriting the corresponding squared-kernel series as $(E_4+10E_2-11)/360$ and evaluating only the resulting linear combinations of $E_2$ and $E_4$ at $i/2$, $i$, and $2i$.

math.NT↗

arXiv2026

A guide to tropical modifications

This paper surveys {\it tropical modifications}, a notion that has become part of the folklore of tropical geometry. Tropical modifications are used in tropical intersection theory, tropical Hodge theory, and the study of singularities. They admit interpretations in several contexts, including hyperbolic geometry, Berkovich spaces, and non-standard analysis. The goal of the paper is to present several points of view, give references, and illustrate the usefulness of tropical modifications. We assume that the reader has already encountered tropical modifications and wants to understand them better. The paper also contains two new contributions: an obstruction to the realizability of non-transversal intersections and a tropical version of Weil's reciprocity law.

math.AG↗

arXiv2026

Residues of a tropical zeta function for convex domains

We define an $\operatorname{SL}_n(\mathbb{Z})$-invariant tropical zeta function of a convex domain. In dimension 2 it admits boundary Dirichlet-series representation with summands indexed by Farey pairs. For $C^3$ strictly convex domains, it extends meromorphically to $\Re(s)>3/5$, holomorphic there except for a simple pole at $s=2/3$, with residue universally proportional to equiaffine perimeter. A Tauberian argument yields the $t^{1/3}$ wave-front lattice-perimeter asymptotic for $t\rightarrow 0$. In addition, for a special domain $L$, which is a limit shape of lattice polygons in a square, with its tropical zeta function being expressed in terms of Witten SU(3) zeta function, we compute the exact coefficient in the asymptotic expansion of the integer-averaged lattice point counting for the leading term $N^{1/2}$.

math.NT↗

arXiv2026

Sampling-Free Privacy Accounting for Matrix Mechanisms under Random Allocation

We study privacy amplification for differentially private model training with matrix factorization under random allocation (also known as the balls-in-bins model). Recent work by Choquette-Choo et al. (2025) proposes a sampling-based Monte Carlo approach to compute amplification parameters in this setting. However, their guarantees either only hold with some high probability or require random abstention by the mechanism. Furthermore, the required number of samples for ensuring $(ε,δ)$-DP is inversely proportional to $δ$. In contrast, we develop sampling-free bounds based on Rényi divergence and conditional composition. The former is facilitated by a dynamic programming formulation to efficiently compute the bounds. The latter complements it by offering stronger privacy guarantees for small $ε$, where Rényi divergence bounds inherently lead to an over-approximation. Our framework applies to arbitrary banded and non-banded matrices. Through numerical comparisons, we demonstrate the efficacy of our approach across a broad range of matrix mechanisms used in research and practice.

cs.LG↗

arXiv2026

The Directed Abelian Sandpile Model on Cylinders

We study the abelian sandpile model in two dimensions on a directed cylindrical lattice with periodic transverse boundary conditions in the transverse direction and dissipation at one boundary. Recurrent configurations form a finite abelian group, and repeated grain addition at a specific site generates deterministic dynamics on this group. Using Dhar's formulation, the sandpile group is identified with the co-kernel of the reduced directed Laplacian. We show that the group structure admits an exact reduction to a transverse problem, allowing complete determination of its cyclic decomposition. Our results establish a direct connection between the algebraic structure of the sandpile group and the periodicity of the driven dynamics, establishing the manner in which the underlying algebraic structure governs both deterministic and stochastic evolution in directed sandpile.

cond-mat.stat-mech↗

arXiv2026

Population Risk Bounds for Kolmogorov-Arnold Networks Trained by DP-SGD with Correlated Noise

We establish the first population risk bounds for Kolmogorov-Arnold Networks (KANs) trained by mini-batch SGD with gradient clipping, covering non-private SGD as well as differentially private SGD (DP-SGD) with Gaussian perturbations that interpolate between independent and temporally correlated noise. This setting is substantially closer to practice than prior KAN theory along two axes: training is by mini-batch SGD, the standard recipe for modern networks, rather than full-batch gradient descent (GD); and correlated-noise mechanisms have empirically shown a more favorable privacy-utility tradeoff than independent-noise mechanisms. Our results cover the corresponding full-batch GD and independent-noise DP-GD results for KANs by Wang et al. (2026), while yielding sharper fixed-second-layer specializations. The technical core is a new analysis route for correlated-noise DP training in the non-convex regime. Temporal dependence breaks the conditional-centering structure underlying standard one-step SGD arguments, and the projection step obstructs the exact cancellation structure of correlated perturbations. We address these difficulties through an auxiliary unprojected dynamics, a shifted iterate that absorbs the current noise perturbation, and a high-probability bootstrap certifying projection inactivity. Combining this optimization analysis with a stability-based generalization argument yields the stated population risk bounds. To the best of our knowledge, this is the first optimization and population risk analysis of a correlated-noise mechanism for DP training beyond convex learning, in particular for neural networks.

cs.LG↗

arXiv2026

Error terms for continued fractions of $e^{1/s}$ and $\sqrt{\frac{v}{u}}\tanh\!\Bigl(\frac{1}{\sqrt{uv}}\Bigr)$

Many classical identities arise from nothing more mysterious than looking at the same object in two different ways. A number, a function, or a combinatorial object may admit several natural decompositions, and by disassembling it in one way and reassembling it in another, we often obtain unexpected corollaries. Telescoping sums provide a particularly vivid incarnation of this principle: by arranging terms so that successive contributions cancel, one performs a conceptual ``cut-and-paste'' that often admits a clean geometric interpretation. Generating functions offer a complementary perspective. Encoding a problem into a formal power series and then evaluating that series at a prescribed point naturally expresses the same quantity as an infinite (or finite) expansion, and equating these representations yields a wealth of identities. For example, for a real number $α$ given by its continued fraction expansion $α= [a_0, a_1,a_2,\dots]$, with convergents $p_n/q_n$ and error terms $E_n := p_n - αq_n$, one can obtain ``additive'' decompositions of the form $\sum_{n\ge-1} a_{n+1}\,\lvert E_n\rvert \;=\; α+ 1$, $\sum_{n\ge-1} a_{n+1}\,E_n^{2} \;=\; α$. Thus $α$ and $α+1$ themselves appear as weighted sums of the local approximation errors of their convergents. In this note we explore what such decompositions yield in two explicit cases: the continued fraction \[ e^{1/s} = [1;\,{\overline{(2k-1)s-1,1,1}}]_{k=1}^{\infty} \] and the continued fraction \[ \frac{s}{u}\tanh\!\Bigl(\frac{1}{s}\Bigr) = [\,0;\,\overline{(4k-3)u,\,(4k-1)\tfrac{s^{2}}{u}}\,]_{k=1}^{\infty} \]

math.GM↗

arXiv2026

Continued fractions, determinant expressions, and identities

In this paper, we clarified the relationship between continued fractions, determinants, and identities, making it easier to apply these methods systematically in other settings. In particular, we studied finite continued fractions from the perspective of incomplete numbers (restricted or associated numbers) and also explored their relationships with determinant representations and identities. Most of the new results in this paper concern $q$-analogues of special numbers, whereas the classical cases mainly serve to illustrate and unify the general framework. The framework developed here is flexible and allows one to derive continued fractions, determinant formulas, and coefficient identities in a uniform way for several new $q$-families, and it is expected to be applicable to other families of special numbers, as well.

math.GM↗

arXiv2026

Evaluating lattice sums via telescoping on $SL_+(2,\mathbb Z)$: a short proof of $\sum \frac{1}{|x|^2|y|^2|x+y|^2}=\fracπ{4}$ and Zagier's identity

We study lattice sums $\sum \frac{1}{(\|x\|\|y\|\|x+y\|)^s}$ taken over $SL_+(2,\mathbb Z)$, i.e.\ the set of pairs $(x,y)$ of primitive lattice vectors in $\mathbb Z_{\geq 0}^2$ with $\det(x, y) = 1$. We prove convergence of these and similar (determinant weighted) sums and introduce a new telescoping method on $SL_+(2,\mathbb Z)$ that yields, in particular, $$\sum_{(x,y)\in SL_+(2,\mathbb Z)} \frac{1}{\|x\|^2\,\|y\|^2\,\|x+y\|^2}=\fracπ{4},$$ and a short proof of Zagier's identity $D_{1,1,1}=2E(z,3)+π^3ζ(3)$.

math.NT↗

arXiv2025

Evaluation of lattice sums via telescoping over topographs

Topographs, introduced by Conway in 1997, are infinite trivalent planar trees used to visualize the values of binary quadratic forms. In this work, we study series whose terms are indexed by the vertices of a topograph and show that they can be evaluated using telescoping sums over its edges. Our technique provides arithmetic proofs for modular graph function identities arising in string theory, yields alternative derivations of Hurwitz-style class number formulas, and provides a unified framework for well-known Mordell-Tornheim series and Hata's series for the Euler constant $γ$. Our theorems are of the following spirit: we cut a topograph along an edge (called the root) into two parts, and then sum $\frac{1}{rst}$ (the reciprocal of the product of labels on regions adjacent to a vertex) over all vertices of one part. We prove that such a sum is equal to an explicit expression depending only on the root and the discriminant of the topograph.

math.NT↗

arXiv2025

A half-shift reflection identity for the digamma function

We prove the identity \[ 2W_1(x) + \log 4 + ψ\left(\tfrac{1}{2} + x\right) + ψ\left(\tfrac{3}{2} - x\right) = 0, \] where $ψ$ is the digamma function and \[ W_1(x) = 2\int_0^\infty \Re\left( \frac{y}{(y^2+1)(e^{π(y+2ix)} - 1)} \right) dy. \] The identity was first conjectured while studying class number $h(D)$ for $D=m^2$ from two complementary perspectives. Our proof, however, is purely analytic: we compute cosine-series expansions of both sides, expressed in terms of the cosine integral Ci$(z)$. Using the above identity and Möbius inversion we find an elementary formula for $$\sum_{\substack{1\le r<m\\ (r,m)=1}} W_1\!\left(\frac{r}{m}\right).$$

math.NT↗