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On lattice point counting in Δ-modular polyhedra [article]

D.V. Gribanov, N.Yu. Zolotykh
2021 arXiv   pre-print
And let all rank order minors of A be bounded by Δ in absolute values.  ...  In particular, d = n for the case (i) and d = n-k for the case (ii).  ...  Note on counting in polytopes defined by convex hulls In this subsection we prove an analogues theorem to Theorem 2 that is stated for polyhedra defined by a convex hull of points.  ... 
arXiv:2010.05768v4 fatcat:yolmhcffrvehvfop4nzm743724

Hypergeometric Galois Actions [article]

Muhammed Uludag, Ismail Saglam
2015 arXiv   pre-print
We outline a project to study the Galois action on a class of modular graphs (special type of dessins) which arise as the dual graphs of the sphere triangulations of non-negative curvature, classified  ...  Because of their connections to hypergeometric functions, there is a hope that these graphs will render themselves to explicit calculation for a study of Galois action on them, unlike the case of a general  ...  We are thankful to Athanase Papadopoulos for inviting us to publish in this volume. Both authors were funded by the grant TUBITAK-110T690.  ... 
arXiv:1509.07820v2 fatcat:xbm63ts2jfc33k4cpz334n57ge

On generalizations of network design problems with degree bounds

Nikhil Bansal, Rohit Khandekar, Jochen Könemann, Viswanath Nagarajan, Britta Peis
2012 Mathematical programming  
tree), and (2) by incorporating 'degree bounds' in other combinatorial optimization problems such as matroid intersection and lattice polyhedra.  ...  Finally, we introduce the crossing lattice polyhedra problem, and obtain a (1, b + 2∆ − 1) approximation under certain condition.  ...  Acknowledgement: We thank Mohit Singh [32] for the integrality gap for general MCST, and Chandra Chekuri for finding an error in the arborescence result in an earlier version [5] of this paper.  ... 
doi:10.1007/s10107-012-0537-8 fatcat:cyu4vwbwtrfnxcwbpujz5yl7vq

Counting BPS States on the Enriques Calabi-Yau

Albrecht Klemm, Marcos Mariño
2008 Communications in Mathematical Physics  
We verify in this way the heterotic results for the D0-D2 generating functional for low genera and find closed expressions for the topological amplitudes on the total space in terms of modular forms, and  ...  One of them leads to the standard D0-D2 counting amplitudes, and from the other one we obtain information about bound states of D0-D4-D2 branes on the Enriques fibre.  ...  We are specially grateful to Greg Moore for extensive discussions on various issues addressed in this paper, and for a detailed critical reading of the manuscript.  ... 
doi:10.1007/s00220-007-0407-z fatcat:nssyirxtenfz5hzsglls2clim4

On Generalizations of Network Design Problems with Degree Bounds [chapter]

Nikhil Bansal, Rohit Khandekar, Jochen Könemann, Viswanath Nagarajan, Britta Peis
2010 Lecture Notes in Computer Science  
tree), and (2) by incorporating 'degree bounds' in other combinatorial optimization problems such as matroid intersection and lattice polyhedra.  ...  In this paper we extend the scope of the iterative relaxation method in two directions: (1) by handling more complex degree constraints in the minimum spanning tree problem (namely, laminar crossing spanning  ...  Acknowledgement: We thank Mohit Singh [32] for the integrality gap for general MCST, and Chandra Chekuri for finding an error in the arborescence result in an earlier version [5] of this paper.  ... 
doi:10.1007/978-3-642-13036-6_9 fatcat:3keidvlb65fafhwreoikytqd5m

Quantum geometry of elliptic Calabi–Yau manifolds

Albrecht Klemm, Jan Manschot, Thomas Wotschke
2012 Communications in Number Theory and Physics  
T -duality on the fibre implies that the topological string free energy also captures the BPS-invariants of D4-branes wrapping the elliptic fibre and a class in the base.  ...  A holomorphic anomaly equation for the topological string free energy is proposed, which is iterative in the genus expansion as well as in the curve classes in the base.  ...  Also we would like to thank Marco Rauch for collaboration in an initial state of the project. Part of the work of J.  ... 
doi:10.4310/cntp.2012.v6.n4.a5 fatcat:ltw5hqnl3nfstg6ew347z6jx7a

Faster algorithm for counting of the integer points number in Δ-modular polyhedra [article]

D. V. Gribanov, D. S. Malyshev
2022 arXiv   pre-print
Given bounds can be used to obtain faster algorithms for the ILP feasibility problem, and for the problem to count integer points in a simplex or in an unbounded Subset-Sum polytope.  ...  in a d-dimensional polytope P, defined by one of the systems above.  ...  Good surveys on the related ∆-modular ILP problems and parameterised ILP complexity are given in [17, 19, 20, 37] .  ... 
arXiv:2110.01732v4 fatcat:mw55oxqemjhjvngjrzcdxfzcsy

Dynamic enforcement of knowledge-based security policies using probabilistic abstract interpretation

Piotr Mardziel, Stephen Magill, Michael Hicks, Mudhakar Srivatsa
2013 Journal of Computer Security  
We focus on developing probabilistic polyhedra in particular, to support numeric programs.  ...  We also show that, for our benchmarks, restricting constraints to octagons or intervals, rather than full polyhedra, can dramatically improve performance while incurring little to no loss in precision.  ...  Acknowledgments The authors are grateful to Jeff Foster, Nataliya Guts, the CSF'11 anonymous referees, and especially Boris Köpf for their helpful comments on drafts of this paper.  ... 
doi:10.3233/jcs-130469 fatcat:xujrukivkfhypbagdrcpyyltsi

The Singular Theta Correspondence, Lorentzian Lattices and Borcherds-Kac-Moody Algebras [article]

Alex Barnard
2003 arXiv   pre-print
This dissertation answers some of the questions raised in Borcherds' papers on Moonshine and Lorentzian reflection groups.  ...  In the case of elementary lattices, we show that these vector-valued forms can be obtained by inducing scalar-valued forms.  ...  It has been a pleasure to work with someone that has such a unique perspective on mathematics.  ... 
arXiv:math/0307102v1 fatcat:mf62iwuqfvdxtbussukzmh5xmy

A non-associative incidence near-ring with a generalized Möbius function [article]

John Johnson, Max Wakefield
2022 arXiv   pre-print
In particular we prove analogues of Phillip Hall's Theorem on the M\"obius function as an alternating sum of chain counts, Weisner's theorem, and Rota's Crosscut Theorem.  ...  Using this generalized M\"obius function we define analogues of the characteristic polynomial and M\"obius polynomials for ranked lattices.  ...  Could there be some chromatic generalization for the generalized polynomials or some lattice point or finite field counting formula for these polynomials (like [7] or [6] )?  ... 
arXiv:2204.07158v1 fatcat:owls7rp6ozhtjewuddg7lvw3ui

A One Parameter Family of Calabi-Yau Manifolds with Attractor Points of Rank Two [article]

Philip Candelas, Xenia de la Ossa, Mohamed Elmi, Duco van Straten
2019 arXiv   pre-print
The values of the periods and their covariant derivatives, at the attractor points, are identified in terms of critical values of the L-functions of the modular groups.  ...  We identify three values of the parameter that give rise to persistent factorisations, one of which is defined over Q, and identify, for all three cases, the associated modular groups.  ...  and hospitality of KIAS in 2018.  ... 
arXiv:1912.06146v1 fatcat:bnkfrcqngzfehcej6bp44afbqa

Calabi-Yau 4-folds and toric fibrations

Maximilian Kreuzer, Harald Skarke
1998 Journal of Geometry and Physics  
Fibrations of k-folds, including the elliptic case, manifest themselves in the N lattice in the following simple way: The polyhedron corresponding to the fiber is a subpolyhedron of that corresponding  ...  We compute all Hodge numbers, using Batyrev's formulas (derived by toric methods) for the first and Vafa's fomulas (obtained by counting of Ramond ground states in N=2 LG models) for the latter class,  ...  This work is supported in part by the Austrian Research Funds FWF (Schrödinger fellowship J012328-PHY) and by the Austrian National Bank under grant Nr. 5674. We would like to thank P. Candelas, C.  ... 
doi:10.1016/s0393-0440(97)00059-4 fatcat:q53dyqrfzvhdbfxamljx3jz4fm

Refined stable pair invariants for E-, M- and [p, q]-strings

Min-xin Huang, Albrecht Klemm, Maximilian Poretschkin
2013 Journal of High Energy Physics  
reduction the generating functions count N=2 Seiberg-Witten gauge theory instantons in four dimensions.  ...  , based on del Pezzo surfaces and elliptic surfaces, in particular the half K3.  ...  We especially thank Sheldon Katz for comments on geometric interpretations of refined BPS states. Parts of MH's work were conducted during his membership at Kavli IPMU.  ... 
doi:10.1007/jhep11(2013)112 fatcat:ajjo44kgjncbxd75hl7efjafyi

A one parameter family of Calabi-Yau manifolds with attractor points of rank two

Philip Candelas, Xenia de la Ossa, Mohamed Elmi, Duco van Straten
2020 Journal of High Energy Physics  
The values of the periods and their covariant derivatives, at the attractor points, are identified in terms of critical values of the L-functions of the modular groups.  ...  We identify three values of the parameter that give rise to persistent factorisations, one of which is defined over ℚ, and identify, for all three cases, the associated modular groups.  ...  This article is distributed under the terms of the Creative Commons Attribution License (CC-BY 4.0), which permits any use, distribution and reproduction in any medium, provided the original author(s)  ... 
doi:10.1007/jhep10(2020)202 fatcat:xqm4yrrldvgbtomxl47jnw7dmm

Modular Forms as Classification Invariants of 4D N=2 Heterotic–IIA Dual Vacua [article]

Yuichi Enoki, Taizan Watari
2019 arXiv   pre-print
It is well-known that one can assign a modular form Φ to such a vacuum by counting perturbative BPS states in Heterotic theory or collecting Noether–Lefschetz numbers associated with the K3-fibration of  ...  In this article, we expand the observations and ideas (using gauge threshold correction) in the literature and formulate a modular form Ψ with full generality for the class of vacua above, which can be  ...  After all, an important point for the embedding trick is to find a decomposition 1 = δ∈N ∨ /Nθ N +δ (τ )h δ (τ ) (215) for some lattice N and a modular form h, not necessarily associated with Niemeier  ... 
arXiv:1911.09934v2 fatcat:h746pyxfxfb4dpkdnhysyndhtq
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