Tootfinder

Interpolations for a quantum Parisi formula in transverse field mean-field spin glass models
C. Itoi, K. Fujiwara, Y. Sakamoto
https://arxiv.org/abs/2506.09419

Interpolations for a quantum Parisi formula in transverse field mean-field spin glass models
A quantum Parisi formula for the transverse field Sherrington-Kirkpatrick (SK) model is proven with an elementary mathematical method. First, a self-overlap corrected quantum model of the transverse field SK model is represented in terms of the Hamiltonian with annealed random interactions. The interpolation given by Guerra and Toninelli is extended to the self-overlap corrected quantum model. It is proven that the infinite-volume limit of the free energy density exists in the operator formalis…

Apparent inconsistency between Streda formula and Hall conductivity in reentrant integer quantum anomalous Hall effect in twisted MoTe$_2$
Yi Huang, Seth Musser, Jihang Zhu, Yang-Zhi Chou, Sankar Das Sarma
https://arxiv.org/abs/2506.10965

Apparent inconsistency between Streda formula and Hall conductivity in reentrant integer quantum anomalous Hall effect in twisted MoTe$_2$
Recent experiments in twisted bilayer MoTe$_2$ (tMoTe$_2$) have uncovered a rich landscape of correlated phases. In this work, we investigate the reentrant integer quantum anomalous Hall (RIQAH) states reported in F. Xu, et. al., arXiv:2504.06972 which displays a notable mismatch between the Hall conductivity measured via transport and that inferred from the Streda formula. We argue that this discrepancy can be explained if the RIQAH state is a quantum Hall bubble phase, analogous to similar we…

That's pretty cool, Roku has #f1 channel. This is different than the app:
https://therokuchannel.roku.com/details/d2564cef9feeec6c58a076d6f6a904…

Roku
Roku provides the simplest way to stream entertainment to your TV. On your terms. With thousands of available channels to choose from.

Recurrence relations and the Christoffel-Darboux formula for elliptic orthogonal polynomials
Harini Desiraju, Sampad Lahiry
https://arxiv.org/abs/2506.09582

Recurrence relations and the Christoffel-Darboux formula for elliptic orthogonal polynomials
In recent years, there has been significant progress in the theory of orthogonal polynomials on algebraic curves, particularly on genus 1 surfaces. In this paper, we focus on elliptic orthogonal polynomials and establish several of their fundamental properties. In particular, we derive general five-term and seven-term recurrence relations, which lead to a Christoffel-Darboux formula and the construction of an associated point process on the A-cycle of the torus. Notably, the recurrence coeffici…

Hall effect in isolated flat-band systems
Raigo Nagashima, Masao Ogata, Naoto Tsuji
https://arxiv.org/abs/2506.09535 https://arxiv.or…

Hall effect in isolated flat-band systems
We study the Hall effect in isolated flat-band systems (i.e., a flat band is separated from other bands) for a weak magnetic field. In a naive semiclassical picture, the Hall conductivity vanishes when dispersive bands are unoccupied since there is no mobile carrier. To go beyond the semiclassical picture, we establish a fully quantum mechanical gauge-invariant formula for the Hall conductivity that can be applied to any lattice models. We apply the formula to a general two-band model with one …

Measuring Semantic Information Production in Generative Diffusion Models
Florian Handke, F\'elix Koulischer, Gabriel Raya, Luca Ambrogioni
https://arxiv.org/abs/2506.10433

Measuring Semantic Information Production in Generative Diffusion Models
It is well known that semantic and structural features of the generated images emerge at different times during the reverse dynamics of diffusion, a phenomenon that has been connected to physical phase transitions in magnets and other materials. In this paper, we introduce a general information-theoretic approach to measure when these class-semantic "decisions" are made during the generative process. By using an online formula for the optimal Bayesian classifier, we estimate the conditional ent…

This https://arxiv.org/abs/2505.03487 has been replaced.
initial toot: https://mastoxiv.page/@arXiv_mat…

Admissible covers and stable maps
Moduli spaces of admissible covers and stable maps of target curves give rise to cycles on $\overline{M}_{g,n}$. We prove a formula relating these cycles. It recovers both the Ekedahl-Lando-Shapiro-Vainshtein formula and the Gromov-Witten/Hurwitz correspondence, providing a cycle-theoretic refinement thereof. The formula is verified computationally in low-genus cases with the help of Johannes Schmitt.

Building on the 90s, statistical n-gram language models, trained on vast text collections, became the backbone of NLP research. They fueled advancements in nearly all NLP techniques of the era, laying the groundwork for today's AI.
F. Jelinek (1997), Statistical Methods for Speech Recognition, MIT Press, Cambridge, MA
#NLP

Generalization of Ramanujan's formula for the sum of half-integer powers of consecutive integers via formal Bernoulli series
Max A. Alekseyev, Rafael Gonzalez, Keryn Loor, Aviad Susman, Cesar Valverde
https://arxiv.org/abs/2506.05961

Generalization of Ramanujan's formula for the sum of half-integer powers of consecutive integers via formal Bernoulli series
Faulhaber's formula expresses the sum of the first $n$ positive integers, each raised to an integer power $p\geq 0$ as a polynomial in $n$ of degree $p+1$. Ramanujan expressed this sum for $p\in\{\frac12,\frac32,\frac52,\frac72\}$ as the sum of a polynomial in $\sqrt{n}$ and a certain infinite series. In the present work, we explore the connection to Bernoulli polynomials, and by generalizing those to formal series, we extend the Ramanujan result to all positive half-integers $p$.

Im using case_when() quite a lot, case_match() is new to me: #rstats

A general vectorised switch() — case_match
This function allows you to vectorise multiple switch() statements. Each case is evaluated sequentially and the first match for each element determines the corresponding value in the output vector. If no cases match, the .default is used. case_match() is an R equivalent of the SQL "simple" CASE WHEN statement. Connection to case_when() While case_when() uses logical expressions on the left-hand side of the formula, case_match() uses values to match against .x with. The following two statement…

Generating Hodges' Graviton MHV Formula with an $Lw_{1 \infty}$ Ward Identity
Alfredo Guevara, Elizabeth Himwich, Noah Miller
https://arxiv.org/abs/2506.05460

Generating Hodges' Graviton MHV Formula with an $Lw_{1+\infty}$ Ward Identity
Hodges' formula expresses the tree-level all-multiplicity Einstein gravity MHV amplitude as a matrix determinant. In this work, we prove that Hodges' determinant is generated by an $Lw_{1+\infty}$ Ward identity on the celestial sphere. The Ward identity takes the form of a recursion relation that has not previously appeared in the literature and is unrelated to BCFW. The proof makes use of the matrix-tree theorem.

Spatially Local Estimates of the Thermal Conductivity of Materials
C. Ugwumadu, A. Gautam, Y. G. Lee, D. A. Drabold
https://arxiv.org/abs/2506.10041 https:…

Spatially Local Estimates of the Thermal Conductivity of Materials
In this paper we describe a spatial decomposition of the thermal conductivity, what we name "site-projected thermal conductivity", a gauge of the thermal conduction activity at each site. The method is based on the Green-Kubo formula and the harmonic approximation, and requires the force-constant and dynamical matrices and of course the structure of a model sitting at an energy minimum. Throughout the paper, we use high quality models previously tested and compared to many experiments. We discu…

I don’t know, asshole, was 1947 worth it? Was close to 80 years of settler colonialism and apartheid worth it? Is the current genocide of the Palestinian people being committed by israel worth it? https://ravenation.club/@welkin7/114652095075122736

Ideals generated by corner-interval minors
Marie Amalore Nambi
https://arxiv.org/abs/2506.09618 https://arxiv.org/pdf/2506.09618

Ideals generated by corner-interval minors
In this article, we study binomial ideals generated by an arbitrary collection of corner-interval $2$-minors of a generic matrix. We determine the minimal prime ideals of such ideals and characterize their radicality in the special case of corner minors. Moreover, we discuss connectivity properties of contingency tables in algebraic statistics. We compute the Hilbert-Poincaré polynomial of the ideal generated by the set of all corner-interval minors and we derive the formula for the regularity…

I realize the interest for this on acitivity pub will be small, but if you are in algorand ecosystem and like #f1, this could be of interest to you: https://www.alphaarcade.com/market/formula

alpha arcade | prediction market on algorand 🌶️
alpha arcade is a decentralized prediction marketplace built on the Algorand blockchain.

The Additive Bachelier model with an application to the oil option market in the Covid period
Roberto Baviera, Michele Domenico Massaria
https://arxiv.org/abs/2506.09760

The Additive Bachelier model with an application to the oil option market in the Covid period
In April 2020, the Chicago Mercantile Exchange temporarily switched the pricing formula for West Texas Intermediate oil market options from the Black model to the Bachelier model. In this context, we introduce an Additive Bachelier model that provides a simple closed-form solution and a good description of the Implied volatility surface. This new Additive model exhibits several notable mathematical and financial properties. It ensures the no-arbitrage condition, a critical requirement in highly…

On Tameness, Measurability and the Independence Property
Lothar Sebastian Krapp, Matthieu Vermeil, Laura Wirth
https://arxiv.org/abs/2506.08733 https://

On Tameness, Measurability and the Independence Property
In the area of Tame Geometry, different model-theoretic tameness conditions are established and their relationships are analyzed. We construct a subfield $K$ of the real numbers that lacks several of such tameness properties. As our main result, we present a first-order formula in the language of rings that defines a non-Borel set in $K$. Moreover, $K$ has the independence property and admits both archimedean and non-archimedean orderings.

Weakly separated self-affine carpets
Bal\'azs B\'ar\'any, Levente David
https://arxiv.org/abs/2506.06851 https://arxiv.or…

Weakly separated self-affine carpets
In this paper, we study the Hausdorff and box-counting dimensions of diagonally aligned self-affine carpets whose projections onto the $x$ and $y$-axes satisfy the weak separation condition. In particular, we will see that the Hausdorff dimension equals the limit of the Barański formula, and the box-counting dimension is the limit of the Feng-Wang formula taken over the $n$th level functions. We also prove various equivalent formulas for the box-counting dimension, and we provide an alternativ…

This https://arxiv.org/abs/2502.19799 has been replaced.
initial toot: https://mastoxiv.page/@arXiv_mat…

New recursion formula for the interior polynomial based on non-expanding sets
The interior polynomial was originally defined for hypergraphs and later shown to coincide with the Ehrhart polynomial of the root polytope of an associated bipartite graph. In previous work, we derived an alternating cycle recursion formula for the interior polynomial. Here, we introduce a new, more transparent recursion formula based on the structure of non-expanding sets. This formula offers a clearer combinatorial interpretation of the interior polynomial and its connection to polyhedral ge…

Israeli military ordered to block aid flotilla carrying Greta Thunberg
The Freedom Flotilla ship is carrying rice, flour, medical supplies, and baby formula to Gaza.
Rashida Tlaib Pens Plea to Trump Admin to Protect Gaza Freedom Flotilla
https://www.politico.eu…

Israeli military ordered to block aid flotilla carrying Greta Thunberg
The Freedom Flotilla Coalition says it is on a mission aimed at delivering supplies to the Gaza Strip.

Ext-group in the category of quantum polynomial functors via the quantum Frobenius twist
Deturck Th\'eo
https://arxiv.org/abs/2506.05827 https://

Ext-group in the category of quantum polynomial functors via the quantum Frobenius twist
We study the effect of a quantum Frobenius twist on Ext-groups in the category of quantum polynomial functors. We use quantum versions of the de Rham and Koszul complexes, and compute their homologies. We use them to do several Ext-computations, and obtain a formula to compute Ext-groups between two functors obtained via the Frobenius, in characteristic zero or in big enough characteristic. Finally, we make some advancements toward a general formula in arbitrary characteristic.

Nonlinear projection for ballistic correlation functions: a formula in terms of minimal connected covers
Benjamin Doyon
https://arxiv.org/abs/2506.05266 ht…

Nonlinear projection for ballistic correlation functions: a formula in terms of minimal connected covers
In many-body systems, the dynamics is governed, at large scales of space and time, by the hydrodynamic principle of projection onto the conserved densities admitted by the model. This is formalised as local relaxation of fluctuations in the ballistic macroscopic fluctuation theory, a nonlinear Boltzmann-Gibbs principle. We use it to derive a projection formula, expressing n-point connected correlation functions (cumulants) of generic observables at different space-time points, in terms of those…

“The Freedom Flotilla Coalition (FFC) confirms that its civilian ship, Madleen, carrying humanitarian aid to Gaza, has been attacked/forcibly intercepted by the Israeli military at 3:02 am CET in international waters at 31.95236° N, 32.38880° E. The ship was unlawfully boarded, its unarmed civilian crew abducted, and its life-saving cargo—including baby formula, food and medical supplies—confiscated.”

This https://arxiv.org/abs/2502.20327 has been replaced.
initial toot: https://mastoxiv.page/@arXiv_mat…

Parabolic bundles and the intersection cohomology of moduli spaces of vector bundles on curves
The study of the intersection cohomology of moduli spaces of semistable bundles was initiated by Frances Kirwan in the 1980's. In this paper, we give a complete geometric proof of a recursive formula, which reduces the calculation of the intersection Betti numbers of the moduli spaces of semistable bundles on Riemann surfaces in degree-0 and arbitrary rank to the known formulas of the Betti numbers of the smooth, degree-1 moduli spaces. Our formula was motivated by the work of Mozgovoy and Rein…

This https://arxiv.org/abs/2408.03465 has been replaced.
initial toot: https://mastoxiv.page/@arXiv_csCC_…

Algorithms for the Diverse-k-SAT problem: the geometry of satisfying assignments
Given a $k$-CNF formula and an integer $s$, we study algorithms that obtain $s$ solutions to the formula that are maximally dispersed. For $s=2$, the problem of computing the diameter of a $k$-CNF formula was initiated by Creszenzi and Rossi, who showed strong hardness results even for $k=2$. Assuming SETH, the current best upper bound [Angelsmark and Thapper '04] goes to $4^n$ as $k \rightarrow \infty$. As our first result, we give exact algorithms for using the Fast Fourier Transform and cliq…

This https://arxiv.org/abs/2407.07774 has been replaced.
initial toot: https://mastoxiv.page/@arXiv_mat…

On a Casselman-Shalika type formula for unramified Speh representations
We give a Casselman-Shalika type formula for unramified Speh representations. Our formula computes values of the normalized spherical element of the $(k,c)$ model of a Speh representation at elements of the form $\operatorname{diag}\left(g, I_{(k-1)c}\right)$, where $g \in \mathrm{GL}_c\left(F\right)$ for a non-archimedean local field $F$. The formula expresses these values in terms of modified Hall--Littlewood polynomials evaluated at the Satake parameter attached to the representation. Our pr…

This https://arxiv.org/abs/2504.17781 has been replaced.
initial toot: https://mastoxiv.page/@arXiv_hept…

Thermal Product Formula for Shear Modes
We investigate the validity of the thermal product formula proposed in [arXiv:2304.12339], for the shear channel fluctuations of R-charged black branes in five-dimensional AdS where the shear mode is coupled with charge diffusion mode at non-zero momentum. When these modes are suitably decoupled, we are able to obtain an exact formula for the two point functions of the boundary current and energy-momentum tensor in terms of the quasinormal modes of this channel. This exact formula is a simple m…

Green functions and a positive mass theorem for asymptotically hyperbolic $3$-manifolds
Klaus Kroencke, Francesca Oronzio, Alan Pinoy
https://arxiv.org/abs/2506.07108

Green functions and a positive mass theorem for asymptotically hyperbolic $3$-manifolds
We prove a new positive mass theorem for three-dimensional manifolds which are asymptotically hyperboloidal of order greater than $1$. The mass quantity under consideration is the volume-renormalized mass recently introduced in a paper by Dahl, McCormick and the first author. The proof is based on a monotonicity formula holding along the level sets of the Green function for the Laplace operator centered at an arbitrary point. In order for this argument to work out, we require that the second ho…

This https://arxiv.org/abs/2407.09881 has been replaced.
initial toot: https://mastoxiv.page/@arXiv_mat…

On the genera of symmetric unions of knots
In the study of ribbon knots, Lamm introduced symmetric unions inspired by earlier work of Kinoshita and Terasaka. We show an identity between the twisted Alexander polynomials of a symmetric union and its partial knot. As a corollary, we obtain an inequality concerning their genera. It is known that there exists an epimorphism between their knot groups, and thus our inequality provides a positive answer to an old problem of Jonathan Simon in this case. Our formula also offers a useful conditio…

CNFs and DNFs with Exactly $k$ Solutions
L. Sunil Chandran, Rishikesh Gajjala, Kuldeep S. Meel
https://arxiv.org/abs/2506.07268 https://

CNFs and DNFs with Exactly $k$ Solutions
Model counting is a fundamental problem that consists of determining the number of satisfying assignments for a given Boolean formula. The weighted variant, which computes the weighted sum of satisfying assignments, has extensive applications in probabilistic reasoning, network reliability, statistical physics, and formal verification. A common approach for solving weighted model counting is to reduce it to unweighted model counting, which raises an important question: {\em What is the minimum …

This https://arxiv.org/abs/2411.05441 has been replaced.
initial toot: https://mastoxiv.page/@arXiv_mat…

Sticky diffusions on star graphs : characterization and It{ô} formula
We investigate continuous diffusions on star graphs with sticky behavior at the vertex. These are Markov processes with continuous paths having a positive occupation time at the vertex. We characterize sticky diffusions as time-changed nonsticky diffusions by adapting the classical technique of It{ô} and McKean. We prove a form of It{ô} formula, also known as Freidlin-Sheu formula, for this type of process. As an intermediate step, we also obtain a stochastic differential equation satisfied b…

Computing Witnesses Using the SCAN Algorithm (Extended Preprint)
Fabian Achammer, Stefan Hetzl, Renate A. Schmidt
https://arxiv.org/abs/2506.00163 https://…

Computing Witnesses Using the SCAN Algorithm (Extended Preprint)
Second-order quantifier-elimination is the problem of finding, given a formula with second-order quantifiers, a logically equivalent first-order formula. While such formulas are not computable in general, there are practical algorithms and subclasses with applications throughout computational logic. One of the most prominent algorithms for second-order quantifier elimination is the SCAN algorithm which is based on saturation theorem proving. In this paper we show how the SCAN algorithm on claus…

Transformation formula of Dwork's $p$-adic hypergeometric function
Yusuke Nemoto
https://arxiv.org/abs/2505.24215 https://arxiv.o…

Transformation formula of Dwork's $p$-adic hypergeometric function
In this paper, we give a transformation formula of Dwork's $p$-adic hypergeometric function between $t$ and $t^{-1}$. As an appendix, we introduce a finite analogue of this transformation formula, which implies the special case of the above transformation formula.

This https://arxiv.org/abs/2412.09080 has been replaced.
initial toot: https://mastoxiv.page/@arXiv_mat…

On the number of modes of Gaussian kernel density estimators
We consider the Gaussian kernel density estimator with bandwidth $β^{-\frac12}$ of $n$ iid Gaussian samples. Using the Kac-Rice formula and an Edgeworth expansion, we prove that the expected number of modes on the real line scales as $Θ(\sqrt{β\logβ})$ as $β,n\to\infty$ provided $n^c\lesssim β\lesssim n^{2-c}$ for some constant $c>0$. An impetus behind this investigation is to determine the number of clusters to which Transformers are drawn in a metastable state.

Trigonometric Ratios Can Prove the Pythagorean Theorem
Shoya Kise, Takesa Uehara, Takashi Shinzato
https://arxiv.org/abs/2506.06304 https://

Trigonometric Ratios Can Prove the Pythagorean Theorem
Recent interest in noncircular trigonometric proofs has underscored the need for alternative methodologies. Jackson and Johnson's 2024 study addresses a longstanding gap in the foundations of trigonometric proofs. Inspired by the work of Jackson and Johnson [JJ24], we present three noncircular proofs of the Pythagorean theorem based on trigonometric identities. First, we establish the Pythagorean theorem via an isosceles triangle construction and the tangent double-angle formula. Second, we pre…

Unknottedness of symplectic submanifold fillings
Zhengyi Zhou
https://arxiv.org/abs/2506.06807 https://arxiv.org/pdf/2506.06807

Unknottedness of symplectic submanifold fillings
We show that any symplectic filling of the standard contact submanifold $(\mathbb{S}^{2n-1},ξ_{\mathrm{std}})$ of $(\mathbb{S}^{2n+1},ξ_{\mathrm{std}})$ in $(\mathbb{D}^{n+1},ω_{\mathrm{std}})$ is smoothly unknotted if $n\ge 2$. We also give a self-contained proof of the Siefring intersection formula between punctured holomorphic curves and holomorphic hypersurfaces used in the proof using the $L$-simple setup of Bao-Honda.

Weyl formula improvement for product of Zoll manifolds
Yanfei Wang
https://arxiv.org/abs/2506.02299 https://arxiv.org/pdf/2506.02299

Weyl formula improvement for product of Zoll manifolds
Iosevich and Wyman have proved in ~\cite{IoWy} that the remainder term in classical Weyl law can be improved from $O(λ^{d-1})$ to $o(λ^{d-1})$ in the case of product manifold by using a famous result of Duistermaat and Guillemin. They also showed that we could have polynomial improvement in the special case of Cartesian product of round spheres by reducing the problem to the study of the distribution of weighted integer lattice points. In this paper, we show that we can extend this result to …

This https://arxiv.org/abs/2205.12953 has been replaced.
link: https://scholar.google.com/scholar?q=a

The Blowup Formula for the Instanton Part of Vafa-Witten Invariants on Projective Surfaces
We prove a blow-up formula for the generating series of virtual $χ_y$-genera for moduli spaces of sheaves on projective surfaces, which is related to a conjectured formula for topological $χ_y$-genera of Göttsche. Our formula is a refinement of one by Vafa-Witten relating to S-duality. We prove the formula simultaneously in the setting of Gieseker stable sheaves on polarised surfaces and also in the setting of framed sheaves on $\mathbb{P}^2$. The proof is based on the blow-up algorithm of N…

Wreath Generalization of Littlewood Reciprocity
Milo Bechtloff Weising
https://arxiv.org/abs/2506.07727 https://arxiv.org/pdf/2506.07…

Wreath Generalization of Littlewood Reciprocity
We generalize Littlewood's reciprocity formula for the restriction coefficients between $\mathrm{GL}_n(\mathbb{C})$ and the symmetric group of permutation matrices $\mathfrak{S}_n$ to the case of the wreath product groups $μ_m^n\rtimes \mathfrak{S}_n $.

This https://arxiv.org/abs/2304.05796 has been replaced.
initial toot: https://mastoxiv.page/@arXiv_mat…

On diagrams of algebras
We present a proof of the formula (given in Lurie's Higher Algebra) for the operad governing diagrams of operad algebras. We believe that our proof corrects a flaw in the original argument. 2nd version: a corrected proof given.

The full formula for the probability of "success" is:
p = {
1/(2^(-n 1)) if n is negative, or
1 - (1/(2^(n 1))) if n is zero or positive
}
(Both branches have the same value when n is 0, so the behavior is smooth around the origin.)
How can we tweak this?
First, we can introduce fixed success and/or failure chances unaffected by level, with this formula only taking effect if those don't apply. For example, you could do 10% failure, 80% by formula, and 10% success to keep things from being too sure either way even when levels are very high or low. On the other hand, this flattening makes the benefit of extra advantage levels even less exciting.
Second, we could allow for gradations of success/failure, and treat the coin pools I used to explain that math like dice pools a bit. An in-between could require linearly more success flips to achieve the next higher grade of success at each grade. For example, simple success on a crit role might mean dealing 1.5x damage, but if you succeed on 2 of your flips, you get 9/4 damage, or on 4 flips 27/8, or on 7 flips 81/16. In this world, stacking crit levels might be a viable build, and just giving up on armor would be super dangerous. In the particular case I was using this for just now, I can't easily do gradations of success (that's the reason I turned to probabilities in the first place) but I think I'd favor this approach when feasible.
The main innovation here over simple dice pools is how to handle situations where the number of dice should be negative. I'm almost certain it's not a truly novel innovation though, and some RPG fan can point out which system already does this (please actually do this, I'm an RPG nerd too at heart).
I'll leave this with one more tweak we could do: what if the number 2 in the probability equation were 3, or 2/3? I think this has a similar effect to just scaling all the modifiers a bit, but the algebra escapes me in this moment and I'm a bit lazy. In any case, reducing the base of the probability exponent should let you get a few more gradations near 50%, which is probably a good thing, since the default goes from 25% straight to 50% and then to 75% with no integer stops in between.

Determinants of Random Unitary Pencils
Michael T. Jury, George Roman
https://arxiv.org/abs/2506.04400 https://arxiv.org/pdf/2506.0440…

Determinants of Random Unitary Pencils
We investigate determinants of random unitary pencils (with scalar or matrix coefficients), which generalize the characteristic polynomial of a single unitary matrix. In particular we examine moments of such determinants, obtained by integrating against the Haar measure on the unitary group. We obtain an exact formula in the case of scalar coefficients, and conjecture an asymptotic formula in the general case, and prove a special case of the conjecture.

#P is Sandwiched by One and Two #2DNF Calls: Is Subtraction Stronger Than We Thought?
Max Bannach, Erik D. Demaine, Timothy Gomez, Markus Hecher
https://

#P is Sandwiched by One and Two #2DNF Calls: Is Subtraction Stronger Than We Thought?
The canonical class in the realm of counting complexity is #P. It is well known that the problem of counting the models of a propositional formula in disjunctive normal form (#DNF) is complete for #P under Turing reductions. On the other hand, #DNF $\in$ spanL and spanL $\not\subseteq$ #P unless NL = NP. Hence, the class of functions logspace-reducible to #DNF is a strict subset of #P under plausible complexity-theoretic assumptions. By contrast, we show that two calls to a (restricted) #2DNF o…

This https://arxiv.org/abs/2505.23667 has been replaced.
initial toot: https://mastoxiv.page/@arXiv_csAI_…

Fortune: Formula-Driven Reinforcement Learning for Symbolic Table Reasoning in Language Models
Tables are a fundamental structure for organizing and analyzing data, making effective table understanding a critical capability for intelligent systems. While large language models (LMs) demonstrate strong general reasoning abilities, they continue to struggle with accurate numerical or symbolic reasoning over tabular data, especially in complex scenarios. Spreadsheet formulas provide a powerful and expressive medium for representing executable symbolic operations, encoding rich reasoning patt…

This https://arxiv.org/abs/2411.15479 has been replaced.
initial toot: https://mastoxiv.page/@arXiv_mat…

Sparse Polynomial Matrix Optimization
A polynomial matrix inequality is a formula asserting that a polynomial matrix is positive semidefinite. Polynomial matrix optimization concerns minimizing the smallest eigenvalue of a symmetric polynomial matrix subject to a tuple of polynomial matrix inequalities. This work explores the use of sparsity methods in reducing the complexity of sum-of-squares based methods in verifying polynomial matrix inequalities or solving polynomial matrix optimization. In the unconstrained setting, Newton po…

This https://arxiv.org/abs/2505.03447 has been replaced.
initial toot: https://mastoxiv.page/@arXiv_mat…

Asymptotic formula for the sum of a prime and a square-full number in intervals shorter than $X^{1/2}$
Let $R(N)$ be the number of representations of $N$ as a sum of a prime and a square-full number weighted with logarithmic function. In $2024$, the author and Y. Suzuki obtained an asymptotic formula for the sum of $R(N)$ over positive integers $N$ in a short interval ($X$, $X+H$] for $X^{\frac{1}{2}+\varepsilon} \le H < X^{1-\varepsilon}$. In this article, we improve the range of $H$, that is, we prove the same asymptotic formula for $X^{\frac{32-4\sqrt{15}}{49}+\varepsilon} \le H \le X^{1- \va…

A Formula for Connected Bosonic \(n\)-Point Functions for the BKP Hierarchy
Xuhui Zhang, Jian Zhou
https://arxiv.org/abs/2506.00451 https://

A Formula for Connected Bosonic \(n\)-Point Functions for the BKP Hierarchy
We present a formula for the connected \(n\)-point functions of a tau-funtion of the BKP hierarchy by embedding BKP hierarchy into KP hierarchy. This formula is different from the one given by Wang and Yang. We prove that these two formulae are equivalent.

On the Farrell--Tate $K$-theory of $\text{Out}(F_n)$
Naomi Andrew, Irakli Patchkoria
https://arxiv.org/abs/2505.21803 https://arxiv.o…

On the Farrell--Tate $K$-theory of $\text{Out}(F_n)$
Using Lück's Chern character isomorphism we obtain a general formula in terms of centralisers for the $p$-adic Farrell--Tate $K$-theory of any discrete group $G$ with a finite classifying space for proper actions. We apply this formula to $\text{Out}(F_n)$. The case $n=p+1$ turns out to be especially interesting for the following reason: Up to conjugacy there is exactly one order $p$ element in $\text{Out}(F_{p+1})$ which does not lift to an order $p$ element in $\text{Aut}(F_{p+1})$. We compu…

Hydrodynamic noise in one dimension: projected Kubo formula and its vanishing in integrable models
Benjamin Doyon
https://arxiv.org/abs/2506.05279 https://…

Hydrodynamic noise in one dimension: projected Kubo formula and its vanishing in integrable models
Hydrodynamic noise is the Gaussian process that emerges at larges scales of space and time in many-body systems. It arises by the central limit theorem applied to local microcanonical averages, representing the degrees of freedom that have been forgotten when projecting coarse-grained observables onto conserved quantities. It comes with "bare" diffusion terms. In one dimension of space, nonlinearities of the hydrodynamic equation are relevant (from a renormalisation perspective), usually giving…

Analytic Uniqueness of Ball Volume Interpolation: Categorical Invariance and Universal Characterization
Andreu Ballus Santacana
https://arxiv.org/abs/2506.06885

Analytic Uniqueness of Ball Volume Interpolation: Categorical Invariance and Universal Characterization
We show that the classical volume formula for the unit $x$-ball, \[ V_x = \frac{π^{x/2}}{Γ(x/2+1)}, \] can be characterized as the \emph{unique} analytic continuation of Haar measure normalization and unit ball volumes for $\mathrm{O}(n)$, under principles of categorical invariance and normalization at integer dimensions. We generalize this result to the unitary and symplectic cases, formalize invariance using categorical language, and construct explicit categorical examples with functorial d…

The complexity of deciding characteristic formulae modulo nested simulation
Luca Aceto, Antonis Achilleos, Aggeliki Chalki, Anna Ingolfsdottir
https://arxiv.org/abs/2505.22277

The complexity of deciding characteristic formulae modulo nested simulation
This paper studies the complexity of determining whether a formula in the modal logics characterizing the nested-simulation semantics is characteristic for some process, which is equivalent to determining whether the formula is satisfiable and prime. The main results are that the problem of determining whether a formula is prime in the modal logic characterizing the 2-nested-simulation preorder is CoNP-complete and is PSPACE-complete in the case of the n-nested-simulation preorder, when n>= 3. …

This https://arxiv.org/abs/2504.04209 has been replaced.
initial toot: https://mastoxiv.page/@arXiv_hepp…

Axion Superradiance in Dipole Magnetic Fields of Pulsars
We consider constraints on the axion-photon coupling by superradiance due to a plasma instability in the magnetospheres of millisecond pulsars. We compute the growth rate of a superradiant axion cloud in a dipole magnetic field, and give a semi-analytical formula for the superradiance rate for the lowest state. By requiring the associated instability time to be longer than the characteristic age of the supermassive black-widow millisecond pulsar PSR J0952-0607, we examine the pulsar-timing arra…

Can Gravitational Wave Data Shed Light on Dark Matter Particles ?
Parthasarathi Majumdar
https://arxiv.org/abs/2506.04905 https://arx…

Can Gravitational Wave Data Shed Light on Dark Matter Particles ?
Gravitational wave (GW) data from observed binary black hole coalescences (BBHC), proven to validate the Hawking Area Theorem (HAT) for black hole horizons, has been demonstrated to unambiguously pick theoretically computed logarithmic corrections to the Bekenstein-Hawking Area Formula, which have a {\it negative} coefficient, when combined with the Generalized Second Law of thermodynamics. We propose a composite, `hybrid' approach to quantum gravity black hole entropy calculation, additively c…

This https://arxiv.org/abs/2502.13067 has been replaced.
initial toot: https://mastoxiv.page/@arXiv_mat…

Spectral geometry of the curl operator on smoothly bounded domains
We show that the spectrum of the curl operator on a generic smoothly bounded domain in three-dimensional Euclidean space consists of simple eigenvalues. The main new ingredient in our proof is a formula for the variation of curl eigenvalues under a perturbation of the domain, reminiscent of Hadamard's formula for the variation of Laplace eigenvalues under Dirichlet boundary conditions. As another application of this variational formula, we simplify the derivation of a well-known necessary condi…

Effective equidistribution of translates of tori in arithmetic homogeneous spaces and applications
Pratyush Sarkar
https://arxiv.org/abs/2506.06948 https:/…

Effective equidistribution of translates of tori in arithmetic homogeneous spaces and applications
Let $Γ< G$ be an arithmetic lattice in a noncompact connected semisimple real algebraic group. For $G$ being one of $\operatorname{SO}(n, 1)^\circ$ for $n \geq 2$, $\operatorname{SL}_2(\mathbb R) \times \operatorname{SL}_2(\mathbb R)$, $\operatorname{SL}_3(\mathbb R)$, $\operatorname{SU}(2, 1)$, $\operatorname{Sp}_4(\mathbb R)$, we prove effective equidistribution of large translates of tori in $G/Γ$. As an application, we obtain an asymptotic counting formula with a power saving error term f…

This https://arxiv.org/abs/2212.11928 has been replaced.
link: https://scholar.google.com/scholar?q=a

The Gauss formulas for Laplacians on submanifolds
There are several types of Laplacians of a vector field on a Riemannian manifold. These include the Bochner and the Hodge Laplacian. The Gauss formula for the Levi-Civita connection relates the extrinsic connection to the intrinsic connection. We extend the Gauss formula for the connection to formulas for the different types of Laplacians of a vector field on a submanifold of any codimension $k\geq 1$. In the process, we derive a Gauss formula for the Ricci operator, formulas for the divergence…

This https://arxiv.org/abs/2412.03290 has been replaced.
initial toot: https://mastoxiv.page/@arXiv_hept…

Holographic reconstruction for AdS Wilson line networks and scalar Witten diagrams
We find a holographic reconstruction formula for gravitational Wilson line network operators in AdS$_2$ evaluated between Ishibashi states of the algebra $sl(2,\mathbb{R})$. It is given in integral form where the integrand is the global conformal block multiplied by a smearing function which is the product of the scalar bulk-to-boundary propagators. The integral can be explicitly calculated as multidimensional series of which arguments are rational functions of endpoint coordinates. In the case…

This https://arxiv.org/abs/2409.02192 has been replaced.
initial toot: https://mastoxiv.page/@arXiv_mat…

A note on cables and the involutive concordance invariants
We prove a formula for the involutive concordance invariants of the cabled knots in terms of that of the companion knot and the pattern knot. As a consequence, we show that any iterated cable of a knot with parameters of the form (odd,1) is not smoothly slice as long as either of the involutive concordance invariants of the knot is nonzero. Our formula also gives new bounds for the unknotting number of a cabled knot, which are sometimes stronger than other known bounds coming from knot Floer ho…

This https://arxiv.org/abs/2409.01198 has been replaced.
initial toot: https://mastoxiv.page/@arXiv_csRO_…

Direct Kinematics, Inverse Kinematics, and Motion Planning of 1-DoF Rational Linkages
This study presents a set of algorithms that deal with trajectory planning of rational single-loop mechanisms with one degree of freedom (DoF). Benefiting from a dual quaternion representation of a rational motion, a formula for direct (forward) kinematics, a numerical inverse kinematics algorithm, and the generation of a driving-joint trajectory are provided. A novel approach using the Gauss-Newton search for the one-parameter inverse kinematics problem is presented. Additionally, a method for…

$p$-Modulus on radially symmetric trees
Prem Raj Prasain
https://arxiv.org/abs/2506.07377 https://arxiv.org/pdf/2506.07377

$p$-Modulus on radially symmetric trees
In this paper, we establish the theory of $p$-modulus of a family of infinite paths on an infinite-rooted tree and then explore its interpretation and properties. One key result is the formulation of $p$-modulus on the infinite tree as a limit of $p$-modulus on truncated trees, with a formula given in terms of a series. Analogous to the existing theory for finite graphs, the $1$-modulus of a family of descending paths in an infinite tree is related to the minimum cut problem, the $2$-modulus is…

On the heat kernel of a Cayley graph of $\operatorname{PSL}_2\mathbb{Z}$
Anders Karlsson, Kamila Kashaeva
https://arxiv.org/abs/2506.02340 https://

On the heat kernel of a Cayley graph of $\operatorname{PSL}_2\mathbb{Z}$
In this paper, we obtain an explicit formula for the heat kernel on the infinite Cayley graph of the modular group $\operatorname{PSL}_2\mathbb{Z}$, given by the presentation $\langle a,b\mid a^2=1, b^3=1\rangle$. Our approach extends the method of Chung--Yau in~\cite{MR1667452} by observing that the Cayley graph strongly and regularly covers a weighted infinite line. We solve the spectral problem on this line to obtain an integral expression for its heat kernel, and then lift this to the Cayle…

Multidimensional tilings and MSO logic
R\'emi Pallen, Ilkka T\"orm\"a
https://arxiv.org/abs/2505.17699 https://arxiv.or…

Multidimensional tilings and MSO logic
We define sets of coulourings of the infinite discrete plane using monadic second order (MSO) formulas. We determine the complexity of deciding whether such a formula defines a subshift, parametrized on the quantifier alternation complexity of the formula. We also study the complexities of languages of MSO-definable sets, giving either an exact classification or upper and lower bounds for each quantifier alternation class.

Raabe's Formula For Gamma Function Via Riemann-Liouville Fractional Integrals And Generalized Glaisher Constants
Efe G\"urel
https://arxiv.org/abs/2505.22666

Raabe's Formula For Gamma Function Via Riemann-Liouville Fractional Integrals And Generalized Glaisher Constants
In this paper, we prove Raabe-type integral formulas for gamma function via left and right sided Riemann-Liouville fractional integrals. As corollaries, we give the left and right sided repeated integration formulas for the log-gamma and related functions. The relationship between the generalized Glaisher constants and aforementioned integrals are investigated.

This https://arxiv.org/abs/2505.03447 has been replaced.
initial toot: https://mastoxiv.page/@arXiv_mat…

Asymptotic formula for the sum of a prime and a square-full number in intervals shorter than $X^{1/2}$
Let $R(N)$ be the number of representations of $N$ as a sum of a prime and a square-full number weighted with logarithmic function. In $2024$, the author and Y. Suzuki obtained an asymptotic formula for the sum of $R(N)$ over positive integers $N$ in a short interval ($X$, $X+H$] for $X^{\frac{1}{2}+\varepsilon} \le H < X^{1-\varepsilon}$. In this article, we improve the range of $H$, that is, we prove the same asymptotic formula for $X^{\frac{32-4\sqrt{15}}{49}+\varepsilon} \le H \le X^{1- \va…

This https://arxiv.org/abs/2504.01359 has been replaced.
initial toot: https://mastoxiv.page/@arXiv_mat…

Monogenic functions over real alternative *-algebras: fundamental results and applications
The concept of monogenic functions over real alternative $\ast$-algebras has recently been introduced to unify several classical monogenic (or regular) functions theories in hypercomplex analysis, including quaternionic, octonionic, and Clifford analysis. This paper explores the fundamental properties of these monogenic functions, focusing on the Cauchy-Pompeiu integral formula and Taylor series expansion in hypercomplex subspaces, among which the non-commutativity and especially non-associativ…

Stochastic portfolio theory with price impact
David Itkin
https://arxiv.org/abs/2506.07993 https://arxiv.org/pdf/2506.07993

Stochastic portfolio theory with price impact
We develop a framework for stochastic portfolio theory (SPT), which incorporates modern nonlinear price impact and impact decay models. Our main result derives the celebrated master formula for (additive) functional generation of trading strategies in a general high-dimensional market model with price impact. We also develop formulas for an investor's relative wealth with respect to the market portfolio, conditions that guarantee positive observed market prices and derive a stochastic different…

General monotone formula for homogeneous $k$-Hessian equation in the exterior domain and its applications
Jiabin Yin, Xingjian Zhou
https://arxiv.org/abs/2506.01434

General monotone formula for homogeneous $k$-Hessian equation in the exterior domain and its applications
In this paper, we deal with an overdetermined problem for the $k$-Hessian equation ($1\leq k<\frac n2$) in the exterior domain and prove the corresponding ball characterizations. Since that Weinberger type approach seems to fail to solve the problem, we give a new perspective to solve exterior overdetermined problem by combining two integral identities and geometric inequalities. In particular, when $k=1$, we give an alternative proof for Brandolini-Nitsch-Salani's results \cite{BNS}. Meanwhile…

Proof of a conjecture by Starr and log-concavity for random commuting permutations
Abdelmalek Abdesselam
https://arxiv.org/abs/2506.06894 https://

Proof of a conjecture by Starr and log-concavity for random commuting permutations
We prove a conjecture by Shannon Starr regarding the asymptotics for the number of tuples of commuting permutations with given number of joint orbits. These numbers generalize unsigned Stirling numbers of the first kind which count how many single permutations have a given number of cycles. In the case of pairs of permutations, these numbers are related to D'Arcais polynomials and the Nekrasov-Okounkov formula. As a consequence of the above asymptotics, we confirm a log-concavity conjecture in …

This https://arxiv.org/abs/2408.06616 has been replaced.
initial toot: https://mastoxiv.page/@arXiv_mat…

Gromov-Witten invariants in family and quantum cohomology
A moduli space of stable maps to the fibers of a fiber bundle is constructed. The new moduli space is a family version of the classical moduli space of stable maps to a non-singular complex projective variety. The virtual cycle for this moduli space is also constructed, and an analogue of Gromov-Witten invariants is defined. As an application, we recover the formula for the number of rational degree d curves in P3, whose image lies in a plane in P3 (known as planar curves in P3), intersecting r…

This https://arxiv.org/abs/2505.01205 has been replaced.
initial toot: https://mastoxiv.page/@arXiv_mat…

The negative $σ$-moment generating function
For $X$ a pre-$λ$ random variable, we show the $σ$-moment generating function of $-X$ can be obtained from the $σ$-moment generating function of $X$ by applying the composition of the standard and degree flip involutions on symmetric power series. This isometric involution is natural as it preserves the pre-$λ$ ring structure on symmetric power series with pre-$λ$ coefficients, thus this formula provides a simple description of the $σ$-moment generating function of $-X$ whenever the $σ$-…

The analytic bootstrap at finite temperature
Julien Barrat, Deniz N. Bozkurt, Enrico Marchetto, Alessio Miscioscia, Elli Pomoni
https://arxiv.org/abs/2506.06422

The analytic bootstrap at finite temperature
We propose new universal formulae for thermal two-point functions of scalar operators based on their analytic structure, constructed to manifestly satisfy all the bootstrap conditions. We derive a dispersion relation in the complexified time plane, which fixes the correlator up to an additive constant and theory-dependent dynamical information. At non-zero spatial separation we introduce a formula for the thermal two-point function obtained by summing over images of the dispersion relation resu…

$W$-entropy formulas and Langevin deformation on the $L^q$-Wasserstein space over Riemannian manifolds
Rong Lei, Xiang-Dong Li, Yu-Zhao Wang
https://arxiv.org/abs/2506.01279

$W$-entropy formulas and Langevin deformation on the $L^q$-Wasserstein space over Riemannian manifolds
Inspired by Perelman's seminal work on the $W$-entropy formula for the Ricci flow and related works, we prove the $W$-entropy formula and rigidity theorem for the geodesic flow on the $L^q$-Wasserstein space over a complete Riemannian manifold with bounded geometry condition. Then we introduce the Langevin deformation on the $L^q$-Wasserstein space over a complete Riemannian manifold, which interpolates between the $p$-Laplacian heat equation and the geodesic flow on the $L^q$-Wasserstein space…

This https://arxiv.org/abs/2205.12798 has been replaced.
link: https://scholar.google.com/scholar?q=a

An involutive dual knot surgery formula
We prove an involutive analog of the dual knot surgery formula of Eftekhary and Hedden-Levine. We also compute a small model for the local equivalence class of the involutive dual knot complex.

Changing Bases with Pipe Dream Combinatorics
Anna Weigandt
https://arxiv.org/abs/2506.07306 https://arxiv.org/pdf/2506.07306

Changing Bases with Pipe Dream Combinatorics
Lascoux and Schützenberger introduced Schubert and Grothendieck polynomials to study the cohomology and K-theory of the complete flag variety. We present explicit combinatorial rules for expressing Grothendieck polynomials in the basis of Schubert polynomials, and vice versa, using the bumpless pipe dreams (BPDs) of Lam, Lee, and Shimozono. A key advantage of BPDs is that they are naturally back stable, which allows us to give a combinatorial formula for expanding back stable Grothendieck poly…

On Top-Down Pseudo-Boolean Model Counting
Suwei Yang, Yong Lai, Kuldeep S. Meel
https://arxiv.org/abs/2506.05232 https://arxiv.org/pd…

On Top-Down Pseudo-Boolean Model Counting
Pseudo-Boolean model counting involves computing the number of satisfying assignments of a given pseudo-Boolean (PB) formula. In recent years, PB model counting has seen increased interest partly owing to the succinctness of PB formulas over typical propositional Boolean formulas in conjunctive normal form (CNF) at describing problem constraints. In particular, the research community has developed tools to tackle exact PB model counting. These recently developed counters follow one of the two e…

The Schur multiplier norm and its dual norm
Erik Christensen
https://arxiv.org/abs/2505.24670 https://arxiv.org/pdf/2505.24670…

The Schur multiplier norm and its dual norm
We present a formula for the Schur multiplier norm of a complex self-adjoint matrix, and a formula for the norm, which is dual to the Schur multiplier norm, of a self-adjoint matrix. For a complex self-adjoint $n \times n $ matrix $X$ we show that its Schur multiplier norm is determined by $$ \|X\|_S = \min \{\, \|\mathrm{diag}(P)\|_\infty \, :\, - P \leq X \leq P \, \}.$$ The dual space of $( M_n(\bc), \|.\|_S)$ is $(M_n(\bc), \|.\|_{cbB}).$ For $X=X…

This https://arxiv.org/abs/2412.01612 has been replaced.
initial toot: https://mastoxiv.page/@arXiv_mat…

Iwasawa theory for weighted graphs
Let $p$ be a prime number and let $d$ be a positive integer. In this paper, we generalize Iwasawa theory for graphs initiated by Gonet and Vallières to weighted graphs. In particular, we prove an analogue of Iwasawa's class number formula and that of Kida's formula for compatible systems of $(\mathbb{Z}/p^n\mathbb{Z})^d$-covers of weighted graphs. We also provide numerical examples of characteristic elements and Iwasawa invariants. At the end of this paper, we give an application of the ideas …

Study of Complexity Factor and Stability of Dynamical Systems in $f(G)$ Gravity
M. Zeeshan Gul, W. Ahmad, M. M. M. Nasir, Faisal Javed, Orhan Donmez, Bander Almutairi
https://arxiv.org/abs/2506.03559

Study of Complexity Factor and Stability of Dynamical Systems in $f(G)$ Gravity
In this paper, we evaluate the complexity of the non-static cylindrical geometry with anisotropic matter configuration in the framework of modified Gauss-Bonnet theory. In this perspective, we calculate modified field equations, the C energy formula, and the mass function that helps to understand the astrophysical structures in this modified gravity. Furthermore, we use the Weyl tensor and obtain different structure scalars by orthogonally splitting the Riemann tensor. One of these scalars, $YT…

Dimension of Besicovitch-Eggleston sets for non-autonomous systems with countable symbolic dynamics
Jonny Imbierski, Charlene Kalle
https://arxiv.org/abs/2506.01803

Dimension of Besicovitch-Eggleston sets for non-autonomous systems with countable symbolic dynamics
In this article we derive a formula for the Hausdorff dimension of Besicovitch-Eggleston level sets associated with non-autonomous dynamics constructed from families of countable affine iterated function systems. The formula obtained shows that the universal-lower-bound phenomenon present in the autonomous case studied by Fan et al. (2010) persists in this non-autonomous setting.

The Minkowski problem for the $k$-torsional rigidity
Xia Zhao, Peibiao Zhao
https://arxiv.org/abs/2505.24494 https://arxiv.org/pdf/25…

The Minkowski problem for the $k$-torsional rigidity
P. Salani [Adv. Math., 229 (2012)] introduced the $k$-torsional rigidity associated with a $k$-Hessian equation and obtained the Brunn-Minkowski inequalities $w.r.t.$ the torsional rigidity in $\mathbb{R}^3$. Following this work, we first construct, in the present paper, a Hadamard variational formula for the $k$-torsional rigidity with $1\leq k\leq n-1$, then we can deduce a $k$-torsional measure from the Hadamard variational formula. Based on the $k$-torsional measure, we propose the Minkowsk…

Fourier transforms and Abel-Jacobi theory
Younghan Bae, Sam Molcho, Aaron Pixton
https://arxiv.org/abs/2506.06116 https://arxiv.org/p…

Fourier transforms and Abel-Jacobi theory
We connect Fourier transforms between compactified Jacobians over the moduli space of stable curves and logarithmic Abel-Jacobi theory. As an application, we compute the pushforward of monomials of divisors on compactified Jacobians in terms of the twisted double ramification cycle formula.

Motivic Steenrod operations at the characteristic via infinite ramification
Toni Annala, Elden Elmanto
https://arxiv.org/abs/2506.05585 https://

Motivic Steenrod operations at the characteristic via infinite ramification
We construct motivic power operations on the mod-$p$ motivic cohomology of $\Fb_p$-schemes using a motivic refinement of Nizioł's theorem. The key input is a purity theorem for motivic cohomology established by Levine. Our operations satisfy the expected properties (naturality, Adem relations, and the Cartan formula) for all bidegrees, generalizing previous results of Primozic which were only know along the ``Chow diagonal.'' We offer geometric applications of our construction: 1) an example o…

Hive is PSPACE-Hard
Dani\"el Andel, Benjamin Rin
https://arxiv.org/abs/2506.03492 https://arxiv.org/pdf/2506.03492

Hive is PSPACE-Hard
Hive is an abstract strategy game played on a table with hexagonal pieces. First published in 2001, it was and continues to be highly popular among both casual and competitive players. In this paper, we show that for a suitably generalized version of the game, the computational problem of determining whether a given player in an arbitrary position has a winning strategy is PSPACE-hard. We do this by reduction from a variant of Generalized Geography we call Formula Game Geography.

The Error in a Smooth Weighted Prime Number Formula and Zero-free Regions for the Riemann Zeta Function
Songlin Han
https://arxiv.org/abs/2505.23795 https:…

The Error in a Smooth Weighted Prime Number Formula and Zero-free Regions for the Riemann Zeta Function
We study the error bound for a smooth weighted prime number theorem, and its implication to the zero-free region for the Riemann zeta function using the method of Pintz. We also give an application to the average number of smooth weighted Goldbach representations and generalize the result to the case of smooth weighted average k-Goldbach representations.

This https://arxiv.org/abs/2409.08740 has been replaced.
initial toot: https://mastoxiv.page/@arXiv_mat…

Another look at qualitative properties of eigenvalues using effective Hamiltonians
The goal of this paper is to review several qualitative properties of well-known eigenvalue problems using a different perspective based on the theory of effective Hamiltonians, working exclusively on the Hopf-Cole transform of the equation. We revisit some monotonicity results as well as the derivation of several scaling limits by means of the Donsker-Varadhan formula, and we point out several differences between the case of quadratic Hamiltonians and non-quadratic ones.

Infinitesimal freeness for orthogonally invariant random matrices
Guillaume C\'ebron (Toulouse), James A Mingo (Queen's)
https://arxiv.org/abs/2506.05139

Infinitesimal freeness for orthogonally invariant random matrices
We introduce a new kind of free independence, called real infinitesimal freeness. We show that independent orthogonally invariant with infinitesimal laws are asymptotically real infinitesimally free. We introduce new cumulants, called real infinitesimal cumulants and show that real infinitesimal freeness is equivalent to vanishing of mixed cumulants. We prove the formula for cumulants with products as entries.

This https://arxiv.org/abs/2407.01227 has been replaced.
initial toot: https://mastoxiv.page/@arXiv_mat…

Non-intersecting paths and the determinant of the distance matrix of a tree
We present the first combinatorial proof of the Graham-Pollak Formula for the determinant of the distance matrix of a tree, via sign-reversing involutions and the Lindström-Gessel-Viennot Lemma. Our approach provides a cohesive and unified framework for the understanding of the existing generalizations and $q$-analogues of the Graham-Pollak Formula, and facilitates the derivation of a natural simultaneous generalizations for them.

The second order Huang-Yang approximation to the Fermi thermodynamic pressure
Xuwen Chen, Jiahao Wu, Zhifei Zhang
https://arxiv.org/abs/2505.23136 https://…

The second order Huang-Yang approximation to the Fermi thermodynamic pressure
We consider a dilute Fermi gas in the thermodynamic limit with interaction potential scattering length $\mathfrak{a}_0$ at temperature $T>0$. We prove the 2nd order Huang-Yang approximation for the Fermi pressure of the system, in which there is a 2nd order term carrying the positive temperature efffect.Our formula is valid up to the temperature $T<ρ^{\frac{2}{3}+\frac{1}{6}}$, which is, by scaling, also necessary for the Huang-Yang formula to hold. Here, $T_F\simρ^{\frac{2}{3}}$ is the Fermi…

This https://arxiv.org/abs/2411.11594 has been replaced.
initial toot: https://mastoxiv.page/@arXiv_mat…

Interval Multiplicities of Persistence Modules
For any persistence module $M$ over a finite poset $\mathbf{P}$, and any interval $I$ in $\mathbf{P}$, we give a formula of the multiplicity $d_M(V_I)$ of the interval module $V_I$ in the indecomposable decomposition of $M$ in terms of the ranks of matrices consisting of structure linear maps of the module $M$, which gives a generalization of the corresponding formula for 1-dimensional persistence modules. This makes it possible to compute the maximal interval-decomposable direct summand of $M$…

Cosmetic surgeries on knots in homology spheres and the Casson--Walker invariant
Kazuhiro Ichihara, In Dae Jong
https://arxiv.org/abs/2506.02682 https://…

Cosmetic surgeries on knots in homology spheres and the Casson--Walker invariant
We study cosmetic surgeries on a knot in a homology sphere. Several constraints on knots and surgery slopes to admit such surgeries are given. Our main ingredient is the rational surgery formula of the Casson--Walker invariant for 2-component links in the 3-sphere.

This https://arxiv.org/abs/2505.20078 has been replaced.
initial toot: https://mastoxiv.page/@arXiv_hept…

Notes on Rindler wave packets in Minkowski spacetime
We consider wave packets of a massless scalar field that have well-localized Rindler energy, and examine how their energy appears to a Minkowski observer to see how the classical gravitational blue-shift formula is modified quantum mechanically. We derive, by using the saddle point approximation, an analytic expression for the Minkowski momentum distribution of such Rindler wave packets. We find a universal lower bound on the uncertainty of Minkowski momentum; the momentum distribution can neve…

This https://arxiv.org/abs/2412.12633 has been replaced.
initial toot: https://mastoxiv.page/@arXiv_mat…

Arborescences of Random Covering Graphs
A rooted arborescence of a directed graph is a spanning tree directed towards a particular vertex. A recent work of Chepuri et al. showed that the arborescences of a covering graph of a directed graph G are closely related to the arborescences of G. In this paper, we study the weighted sum of arborescences of a random covering graph and give a formula for the expected value, resolving a conjecture of Chepuri et al.

This https://arxiv.org/abs/2410.10884 has been replaced.
initial toot: https://mastoxiv.page/@arXiv_mat…

On evaluation of the lattice sum $\frac{1}{|x||y||x+y|}$
We study lattice sums $\sum 1/|x||y||x+y|$ and $\sum 1/|x|^2|y|^2|x+y|^2$ taken over the pairs $(x,y)$ of primitive lattice vectors in $\ZZ^2$ that span a parallelogram of area one. We prove convergence of these and similar sums, propose a new method to evaluate $\sum 1/|x|^2|y|^2|x+y|^2$, which, in turn, gives a new and simple proof of Zagier's formula for $D_{1,1,1}$.

Cohomology of hypersurfaces of weighted projective space and the intersection form on $H^2$
Anna-Maria Raukh
https://arxiv.org/abs/2506.04094 https://

Cohomology of hypersurfaces of weighted projective space and the intersection form on $H^2$
Given a hypersurface $i \colon X \hookrightarrow \widetilde{P}^n$ in a weighted projective space, we compute the intersection form on the second cohomology $H^2(X, \mathbb{Z})^{\otimes n-1} \to \mathbb{Z}$ for the purpose of identifying Fano manifolds obtained from smoothing singular Fanos. In the process, we describe the integer cohomology groups $H^k(X, \mathbb{Z})$ for $k

This https://arxiv.org/abs/2410.02270 has been replaced.
initial toot: https://mastoxiv.page/@arXiv_hept…

Beyond the ensemble paradigm in low dimensional quantum gravity: Schwarzian density, quantum chaos and wormhole contributions
Based on periodic orbit theory we address the individual-system versus ensemble interpretation of quantum gravity from a quantum chaos perspective. To this end we show that the spectrum of geodesic motion on high-dimensional hyperbolic manifolds, described by the Selberg trace formula, displays a Schwarzian ($\sinh 2π\sqrt{E}$) mean level density. Due to its chaotic classical limit, this quantum system also shows all universal signatures of quantum chaos. These two properties imply a possible …

This https://arxiv.org/abs/2407.18398 has been replaced.
initial toot: https://mastoxiv.page/@arXiv_mat…

Low-weight codewords in cyclic codes
We introduce a formula for determining the number of codewords of weight 2 in cyclic codes and provide results related to the count of codewords with weight 3. Additionally, we establish a recursive relationship for binary cyclic codes that connects their weight distribution to the number of solutions of associated systems of polynomial equations. This relationship allows for the computation of weight distributions from known solutions of systems of diagonal equations and vice versa, offering a…

This https://arxiv.org/abs/2407.13450 has been replaced.
initial toot: https://mastoxiv.page/@arXiv_mat…

Sparse Nullstellensatz, resultants and determinants of complexes
We refine and extend a result by Tuitman on the supports of a Bezout identity satisfied by a finite sequence of sparse Laurent polynomials without common zeroes in the toric variety associated to their supports. When the number of these polynomials is one more than the dimension of the ambient space, we obtain a formula for computing the sparse resultant as the determinant of a Koszul type complex.

This https://arxiv.org/abs/2407.08881 has been replaced.
initial toot: https://mastoxiv.page/@arXiv_mat…

Newspaces with Nebentypus: An Explicit Dimension Formula and Classification of Trivial Newspaces
Consider $N \geq 1$, $k \geq 2$, and $χ$ a Dirichlet character modulo $N$ such that $χ(-1) = (-1)^k$. For any bound $B$, one can show that $\dim S_k(Γ_0(N),χ) \le B$ for only finitely many triples $(N,k,χ)$. It turns out that this property does not extend to the newspace; there exists an infinite family of triples $(N,k,χ)$ for which $\dim S_k^{\text{new}}(Γ_0(N),χ) = 0$. However, we classify this case entirely. We also show that excluding the infinite family for which $\dim S_k^{\text{…

Additivity of non-acyclicity classes for constructible \'etale sheaves
Jiangnan Xiong
https://arxiv.org/abs/2505.24345 https://ar…

Additivity of non-acyclicity classes for constructible étale sheaves
Using a bivariant version of cohomological correspondences, we establish a categorical trace-like formula for the non-acyclicity classes introduced by Yang and Zhao (arXiv:2209.11086). As an application, we prove the additivity for the non-acyclicity classes.

On Murmurations and Trace Formulas
David Lowry-Duda
https://arxiv.org/abs/2506.01640 https://arxiv.org/pdf/2506.01640

On Murmurations and Trace Formulas
In recent work with Bober, Booker, Lee, Seymour-Howell, and Zubrilina, we proved murmuration behavior for Maass forms in the eigenvalue aspect and for modular forms in the weight aspect. Both used an approach based on the Selberg trace formula. But different trace formulas, including those due to Kuznetsov or Petersson, offer different variations. We examine murmurations from the perspective of different trace formulas and outline several families of $L$-functions where one can likely prove add…