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Journal articles on the topic 'Flat functors'

Author: Grafiati

Published: 10 December 2022

Last updated: 31 July 2025

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1

Crivei, Septimiu. "On Flat Objects of Finitely Accessible Categories." Scientific World Journal 2013 (2013): 1–4. http://dx.doi.org/10.1155/2013/451091.

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Flat objects of a finitely accessible additive category are described in terms of some objects of the associated functor category of , called strongly flat functors. We study closure properties of the class of strongly flat functors, and we use them to deduce the known result that every object of a finitely accessible abelian category has a flat cover.

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2

Hu, Hongde. "Flat functors and free exact categories." Journal of the Australian Mathematical Society. Series A. Pure Mathematics and Statistics 60, no. 2 (1996): 143–56. http://dx.doi.org/10.1017/s1446788700037575.

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AbstractLet C be a small category with weak finite limits, and let Flat(C) be the category of flat functors from C to the category of small sets. We prove that the free exact completion of C is the category of set-valued functors of Flat (C) which preserve small products and filtered colimits. In case C has finite limits, this gives A. Carboni and R. C. Magno's result on the free exact completion of a small category with finite limits.

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3

Hu, Jiangsheng, Huanhuan Li, Yuxian Geng, and Dongdong Zhang. "Frobenius functors and Gorenstein flat dimensions." Communications in Algebra 48, no. 3 (2019): 1257–65. http://dx.doi.org/10.1080/00927872.2019.1677699.

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4

Crivei, Septimiu. "Flat Weakly FP-Injective and FP-Injective Weakly Flat Functors." Results in Mathematics 71, no. 3-4 (2016): 1031–45. http://dx.doi.org/10.1007/s00025-016-0598-8.

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5

Nicholson, W. K., and J. F. Watters. "Morita context functors." Mathematical Proceedings of the Cambridge Philosophical Society 103, no. 3 (1988): 399–408. http://dx.doi.org/10.1017/s0305004100065014.

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AbstractGiven a Morita context (R, V, W, S), there are functors W⊗() and hom (V, ) from R-mod to; S-mod and a natural transformation λ from the first to the second. This has an epi-mono factorization and the intermediate functor we denote by ()° with natural transformations and . The tensor functor is exact if and only if WR is flat, whilst the hom functor is exact if and only if RV is projective. We begin by determining conditions under which ()° is exact; this is Theorem 1.

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6

GUO, SHOUTAO, and XIAOYAN YANG. ""ADJOINT PREENVELOPES AND ADJOINT PRECOVERS IN THE FUNCTOR CATEGORY"." Mathematical Reports 25(75), no. 1 (2022): 85–102. http://dx.doi.org/10.59277/mrar.2023.25.75.1.85.

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"Adjoint preenvelopes and adjoint precovers are defined in the category of functors by replacing the functor Hom with ⊗. We investigate the existence and basic properties of adjoint preenvelopes and adjoint precovers. The F-projective (F-injective, F-flat) functors introduced by Mao are characterized in terms of adjoint preenvelopes and adjoint precovers. We obtain relationships among adjoint preenvelopes, adjoint precovers, preenvelopes and precovers."

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7

Farjoun, Emmanuel Dror, and Jérôme Scherer. "Conditionally flat functors on spaces and groups." Collectanea Mathematica 66, no. 1 (2014): 149–60. http://dx.doi.org/10.1007/s13348-013-0102-7.

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8

Mao, Lixin. "Strongly FP-injective and strongly flat functors." Rendiconti del Seminario Matematico della Università di Padova 135 (2016): 133–49. http://dx.doi.org/10.4171/rsmup/135-7.

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9

Aldrich, Stephen T., Edgar E. Enochs, and Juan A. Lopez Ramos. "Derived Functors of Hom Relative to Flat Covers." Mathematische Nachrichten 242, no. 1 (2002): 17–26. http://dx.doi.org/10.1002/1522-2616(200207)242:1<17::aid-mana17>3.0.co;2-f.

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10

Akatsa, Victor K. "Flat resolvents, minimal injective resolutions and negative torsion functors." Communications in Algebra 20, no. 11 (1992): 3105–18. http://dx.doi.org/10.1080/00927879208824508.

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11

Selvaraj, C., and R. Udhayakumar. "Derived Functors of Hom Relative to n-Flat Covers." Vietnam Journal of Mathematics 43, no. 3 (2014): 571–82. http://dx.doi.org/10.1007/s10013-014-0097-8.

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12

Estrada, Sergio, Marco A. Pérez, and Haiyan Zhu. "Balanced pairs, cotorsion triplets and quiver representations." Proceedings of the Edinburgh Mathematical Society 63, no. 1 (2019): 67–90. http://dx.doi.org/10.1017/s0013091519000270.

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AbstractBalanced pairs appear naturally in the realm of relative homological algebra associated with the balance of right-derived functors of the Hom functor. Cotorsion triplets are a natural source of such pairs. In this paper, we study the connection between balanced pairs and cotorsion triplets by using recent quiver representation techniques. In doing so, we find a new characterization of abelian categories that have enough projectives and injectives in terms of the existence of complete hereditary cotorsion triplets. We also provide a short proof of the lack of balance for derived functor

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13

Zhou, Borong. "A note on Morita context functors." Mathematical Proceedings of the Cambridge Philosophical Society 107, no. 2 (1990): 273–74. http://dx.doi.org/10.1017/s0305004100068547.

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It is claimed in [3], proposition 4 that if a Morita context (R, V, W, S) is such that WVW = W and S/WV is flat as a right S-module, then Vω = 0, ω ∈ W, implies ω = 0. The proof given for this uses proposition 5 of Azumaya[l] in which rings have unity and so is only applicable when the ring S has a unity. Thus proposition 4 in [3] should be amended so that the additional hypothesis that S has a unity is imposed for statement (4) to be equivalent to the first three statements. It is the purpose of this note to give an example to show that the claim in [3] is false in general.

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14

Mao, Lixin. "Notes on several orthogonal classes of flat and FP-injective functors." Mathematical Notes 95, no. 1-2 (2014): 78–90. http://dx.doi.org/10.1134/s0001434614010088.

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15

Fernández-Alonso, Rogelio, and Janeth Magaña. "Galois connections between lattices of preradicals induced by ring epimorphisms." Journal of Algebra and Its Applications 19, no. 03 (2019): 2050045. http://dx.doi.org/10.1142/s0219498820500450.

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We continue the study of Galois connections between the lattices of preradicals of two rings [Formula: see text] and [Formula: see text] induced by an adjoint pair of functors between the categories [Formula: see text]-Mod and [Formula: see text]-Mod. In this paper, we focus on the functor triple induced by any ring homomorphism [Formula: see text], and particularly when it is a ring epimorphism. We give additional results when the epimorphism is flat and when it is projective.

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Fu, Xianhui, and Meiyuan Ni. "The Cotorsion Pair Cogenerated by the Subcategory of Stable Functors in ((R-mod)op, Ab)." Algebra Colloquium 26, no. 02 (2019): 259–70. http://dx.doi.org/10.1142/s1005386719000208.

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Let R be an associative ring with identity. Denote by ((R-mod)op, Ab) the category consisting of contravariant functors from the category of finitely presented left R-modules R-mod to the category of abelian groups Ab. An object in ((R-mod)op, Ab) is said to be a stable functor if it vanishes on the regular module R. Let [Formula: see text] be the subcategory of stable functors. There are two torsion pairs [Formula: see text] and [Formula: see text], where ℱ1 is the subcategory of ((R-mod)op, Ab) consisting of functors with flat dimension at most 1. In this article, let R be a ring of weakly g

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17

Ishida, Kazuhisa, Yasunari Shidama, and Adam Grabowski. "Definition of Flat Poset and Existence Theorems for Recursive Call." Formalized Mathematics 22, no. 1 (2014): 1–10. http://dx.doi.org/10.2478/forma-2014-0001.

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Summary This text includes the definition and basic notions of product of posets, chain-complete and flat posets, flattening operation, and the existence theorems of recursive call using the flattening operator. First part of the article, devoted to product and flat posets has a purely mathematical quality. Definition 3 allows to construct a flat poset from arbitrary non-empty set [12] in order to provide formal apparatus which eanbles to work with recursive calls within the Mizar langauge. To achieve this we extensively use technical Mizar functors like BaseFunc or RecFunc. The remaining part

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Iyengar, Srikanth B., Joseph Lipman, and Amnon Neeman. "Relation between two twisted inverse image pseudofunctors in duality theory." Compositio Mathematica 151, no. 4 (2014): 735–64. http://dx.doi.org/10.1112/s0010437x14007672.

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Grothendieck duality theory assigns to essentially finite-type maps $f$ of noetherian schemes a pseudofunctor $f^{\times }$ right-adjoint to $\mathsf{R}f_{\ast }$, and a pseudofunctor $f^{!}$ agreeing with $f^{\times }$ when $f$ is proper, but equal to the usual inverse image $f^{\ast }$ when $f$ is étale. We define and study a canonical map from the first pseudofunctor to the second. This map behaves well with respect to flat base change, and is taken to an isomorphism by ‘compactly supported’ versions of standard derived functors. Concrete realizations are described, for instance for maps of

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19

Yang, Xiaoyan, and Li Wang. "Homological invariants over non-positive DG-rings." Journal of Algebra and Its Applications 19, no. 08 (2019): 2050153. http://dx.doi.org/10.1142/s0219498820501534.

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This paper explores various notions of projective, injective and flat dimensions, and the different versions of each dimension are compared to each other. Homological characterizations of these dimensions are provided in terms of bounded resolutions and vanishing of appropriate derived functors over a non-positive DG-ring. Some homological identities related to these homological invariants, such as Auslander–Buchsbaum formula, Bass formula and derived depth formula, are proved over a commutative Noetherian DG-ring.

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20

ZHANG, DONGDONG, and BAIYU OUYANG. "SEMIDUALIZING MODULES AND RELATED MODULES." Journal of Algebra and Its Applications 10, no. 06 (2011): 1261–82. http://dx.doi.org/10.1142/s0219498811005695.

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In this paper, we prove that the Bass class [Formula: see text] with respect to a semidualizing bimodule C contains all FP-injective S-modules. We introduce the definition of C-FP-injective modules, and give some characterizations of right coherent rings in terms of the C-flat S-modules and C-FP-injective S op -modules. We discuss when every S-module has an C-flat preenvelope which is epic (or monic). In addition, we investigate the left and right [Formula: see text]-resolutions of R-modules by left derived functors Ext n(-, -) over a left Noetherian ring S. As applications, some new character

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21

Abellán García, Fernando. "Marked colimits and higher cofinality." Journal of Homotopy and Related Structures 17, no. 1 (2021): 1–22. http://dx.doi.org/10.1007/s40062-021-00296-2.

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AbstractGiven a marked $$\infty $$ ∞ -category $$\mathcal {D}^{\dagger }$$ D † (i.e. an $$\infty $$ ∞ -category equipped with a specified collection of morphisms) and a functor $$F: \mathcal {D}\rightarrow {\mathbb {B}}$$ F : D → B with values in an $$\infty $$ ∞ -bicategory, we define "Equation missing", the marked colimit of F. We provide a definition of weighted colimits in $$\infty $$ ∞ -bicategories when the indexing diagram is an $$\infty $$ ∞ -category and show that they can be computed in terms of marked colimits. In the maximally marked case $$\mathcal {D}^{\sharp }$$ D ♯ , our constr

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22

Asadollahi, Javad, and Shokrollah Salarian. "Complete Cohomologies and Some Homological Invariants." Algebra Colloquium 14, no. 01 (2007): 155–66. http://dx.doi.org/10.1142/s1005386707000156.

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There is a complete cohomology theory developed over a commutative noetherian ring in which injectives take the role of projectives in Vogel's construction of complete cohomology theory. We study the interaction between this complete cohomology, that is referred to as [Formula: see text]-complete cohomology, and Vogel's one and give some sufficient conditions for their equivalence. Using [Formula: see text]-complete functors, we assign a new homological invariant to any finitely generated module over an arbitrary commutative noetherian local ring, that would generalize Auslander's delta invari

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23

Zhang, Dongdong, and Baiyu Ouyang. "On n-Coherent Rings and (n,d)-Injective Modules." Algebra Colloquium 22, no. 02 (2015): 349–60. http://dx.doi.org/10.1142/s1005386715000309.

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Let R be a ring, n, d be fixed non-negative integers, [Formula: see text] the class of (n,d)-injective left R-modules, and [Formula: see text] the class of (n,d)-flat right R-modules. In this paper, we prove that if R is a left n-coherent ring and m ≥ 2, then [Formula: see text] if and only if [Formula: see text], if and only if Ext m+k(M,N) = 0 for all left R-modules M, N and all k ≥ -1, if and only if Ext m-1(M,N) = 0 for all left R-modules M, N. Meanwhile, we prove that if R is a left n-coherent ring, then − ⊗ − is right balanced on [Formula: see text] by [Formula: see text], and investigat

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24

Belas, Robert, Eiko Horikawa, Shin-Ichi Aizawa, and Rooge Suvanasuthi. "Genetic Determinants of Silicibacter sp. TM1040 Motility." Journal of Bacteriology 191, no. 14 (2009): 4502–12. http://dx.doi.org/10.1128/jb.00429-09.

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ABSTRACT Silicibacter sp. TM1040 is a member of the marine Roseobacter clade of Alphaproteobacteria that forms symbioses with unicellular eukaryotic phytoplankton, such as dinoflagellates. The symbiosis is complex and involves a series of steps that physiologically change highly motile bacteria into cells that readily form biofilms on the surface of the host. The initial phases of symbiosis require bacterial motility and chemotaxis that drive the swimming bacteria toward their planktonic host. Cells lacking wild-type motility fail to establish biofilms on host cells and do not produce effectiv

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25

Hug, Daniel, Günter Last, and Wolfgang Weil. "Distance measurements on processes of flats." Advances in Applied Probability 35, no. 1 (2003): 70–95. http://dx.doi.org/10.1239/aap/1046366100.

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Distance measurements are useful tools in stochastic geometry. For a Boolean modelZin ℝd, the classical contact distribution functions allow the estimation of important geometric parameters ofZ. In two previous papers, several types of generalized contact distributions have been investigated and applied to stationary and nonstationary Boolean models. Here, we consider random setsZwhich are generated as the union sets of Poisson processesXofk-flats,k∈ {0, …,d-1}, and study distances from a fixed point or a fixed convex body toZ. In addition, we also consider the distances from a given flat or a

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Hug, Daniel, Günter Last, and Wolfgang Weil. "Distance measurements on processes of flats." Advances in Applied Probability 35, no. 01 (2003): 70–95. http://dx.doi.org/10.1017/s000186780001209x.

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Distance measurements are useful tools in stochastic geometry. For a Boolean modelZin ℝd, the classical contact distribution functions allow the estimation of important geometric parameters ofZ. In two previous papers, several types of generalized contact distributions have been investigated and applied to stationary and nonstationary Boolean models. Here, we consider random setsZwhich are generated as the union sets of Poisson processesXofk-flats,k∈ {0, …,d-1}, and study distances from a fixed point or a fixed convex body toZ. In addition, we also consider the distances from a given flat or a

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27

Kalmokoff, Martin, Patricia Lanthier, Tammy-Lynn Tremblay, et al. "Proteomic Analysis of Campylobacter jejuni 11168 Biofilms Reveals a Role for the Motility Complex in Biofilm Formation." Journal of Bacteriology 188, no. 12 (2006): 4312–20. http://dx.doi.org/10.1128/jb.01975-05.

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ABSTRACT Campylobacter jejuni remains the leading cause of bacterial gastroenteritis in developed countries, and yet little is known concerning the mechanisms by which this fastidious organism survives within its environment. We have demonstrated that C. jejuni 11168 can form biofilms on a variety of surfaces. Proteomic analyses of planktonic and biofilm-grown cells demonstrated differences in protein expression profiles between the two growth modes. Proteins involved in the motility complex, including the flagellins (FlaA, FlaB), the filament cap (FliD), the basal body (FlgG, FlgG2), and the

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Sun, Jixiang, Cheng Tang, Ke Mu, Yanfang Li, Xiangyang Zheng, and Tao Zou. "Tidal Flat Extraction and Analysis in China Based on Multi-Source Remote Sensing Image Collection and MSIC-OA Algorithm." Remote Sensing 16, no. 19 (2024): 3607. http://dx.doi.org/10.3390/rs16193607.

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Tidal flats, a critical part of coastal wetlands, offer unique ecosystem services and functions. However, in China, these areas are under significant threat from industrialization, urbanization, aquaculture expansion, and coastline reconstruction. There is an urgent need for macroscopic, accurate and periodic tidal flat resource data to support the scientific management and development of coastal resources. At present, the lack of macroscopic, accurate and periodic high-resolution tidal flat maps in China greatly limits the spatio-temporal analysis of the dynamic changes of tidal flats in Chin

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Adela Purba and Nelson Siahaan. "Design of Flat With Neighborhood Space With Urban Farming Function In Medan." International Journal of Architecture and Urbanism 5, no. 3 (2021): 387–97. http://dx.doi.org/10.32734/ijau.v5i3.7720.

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In Medan, flats do not utilize neighborhood space as an urban farming area. In general, flats have a high density of occupants, inflexible designs, and the relationship between one door to another door is relatively close so that communal space is needed. Communal space is a space that functions as a place for social interaction activities for its residents. This flat design has green open spaces such as neighborhood space that can be used as communal spaces both socially and economically productively (urban farming) because flats require both functions. The design approach process used in thi

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30

Kozłowski, Wojciech. "∇-flat functions on manifolds." Annales Polonici Mathematici 84, no. 2 (2004): 177–80. http://dx.doi.org/10.4064/ap84-2-7.

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31

BUCATARU, IOAN, and ZOLTÁN MUZSNAY. "FINSLER METRIZABLE ISOTROPIC SPRAYS AND HILBERT’S FOURTH PROBLEM." Journal of the Australian Mathematical Society 97, no. 1 (2014): 27–47. http://dx.doi.org/10.1017/s1446788714000111.

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AbstractIt is well known that a system of homogeneous second-order ordinary differential equations (spray) is necessarily isotropic in order to be metrizable by a Finsler function of scalar flag curvature. In our main result we show that the isotropy condition, together with three other conditions on the Jacobi endomorphism, characterize sprays that are metrizable by Finsler functions of scalar flag curvature. We call these conditions the scalar flag curvature (SFC) test. The proof of the main result provides an algorithm to construct the Finsler function of scalar flag curvature, in the case

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Bulman-Fleming, Sydney. "Pullback-Flat Acts are Strongly Flat." Canadian Mathematical Bulletin 34, no. 4 (1991): 456–61. http://dx.doi.org/10.4153/cmb-1991-073-2.

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AbstractLet 5 be a monoid. A right S-system A is called strongly flat if the functor A ⊗ — (from the category of left S-systems into the category of sets) preserves pullbacksand equalizers. (This concept arises in B. Stenström, Math. Nachr. 48(1971), 315-334 under the name weak flatness). The main result of the present paper is a proof that for A to be strongly flat it is in fact sufficient that A ⊗ — preserve only pullbacks. The approach taken is to develop an "interpolation" condition for pullback-preservation, and then to show its equivalence to Stenström's conditions for strong flatness.

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Gołębiowska, Anna, and Dariusz Prokopowicz. "NATIONAL HOUSING PROGRAM OF THE MAIN STRATEGIC AND PROGRAM DOCUMENT FOR THE CURRENT STATE RESIDENTIAL POLICY OF THE STATE." International Journal of New Economics and Social Sciences 6, no. 2 (2017): 0. http://dx.doi.org/10.5604/01.3001.0010.7644.

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Launched in April 2016, the Family 500 Plus program significantly reduces the scale of poverty in large families, contributes to the growth of consumption and perhaps also to the increase in fertility in Poland. The second key program of social family policy in Poland launched in December 2016 is the Flat Plus program. This program, through the improvement of the housing situation, should fulfill important functions of housing policy in the scope of reducing the social stratification of families in Poland. The main objective of the Flat Plus program is to significantly increase the availabilit

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34

Yen, Jiun Y., Katherine M. Broadway, and Birgit E. Scharf. "Minimum Requirements of Flagellation and Motility for Infection of Agrobacterium sp. Strain H13-3 by Flagellotropic Bacteriophage 7-7-1." Applied and Environmental Microbiology 78, no. 20 (2012): 7216–22. http://dx.doi.org/10.1128/aem.01082-12.

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ABSTRACTThe flagellotropic phage 7-7-1 specifically adsorbs toAgrobacteriumsp. strain H13-3 (formerlyRhizobium lupiniH13-3) flagella for efficient host infection. TheAgrobacteriumsp. H13-3 flagellum is complex and consists of three flagellin proteins: the primary flagellin FlaA, which is essential for motility, and the secondary flagellins FlaB and FlaD, which have minor functions in motility. Using quantitative infectivity assays, we showed that absence of FlaD had no effect on phage infection, while absence of FlaB resulted in a 2.5-fold increase in infectivity. AflaAdeletion strain, which p

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35

Handayani, Nur, Irwin Irwin, and Muhammad Nurhamsyah. "PERANCANGAN RUMAH SUSUN MAHASISWA UNIVERSITAS TANJUNGPURA DENGAN KONSEP MINIMALIS." JMARS: Jurnal Mosaik Arsitektur 11, no. 2 (2023): 39. http://dx.doi.org/10.26418/jmars.v11i2.59588.

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Universities located in the city center and far from district locations and areas. Not a few students who migrate to continue their education to tertiary level. This condition certainly requires the existence of temporary housing for overseas students during their education period. Pontianak City is one of the city centers where there is a university, namely, Tanjungpura University. One of the temporary residences that is still rare is the existence of special student flats. These student flats can assist students in helping with temporary housing. This student flat is designed with a minimali

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Bulman-Fleming, Sydney, and Kenneth McDowell. "Monoids over which all weakly flat acts are flat." Proceedings of the Edinburgh Mathematical Society 33, no. 2 (1990): 287–98. http://dx.doi.org/10.1017/s0013091500018198.

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If R is a ring with identity and M is a left R-module then it is well known that the following statements are equivalent:(1) M is flat.(2) The functor −⊗M preserves embeddings of right ideals into R.This paper investigates situations in which the analogous statements are equivalent in the context of S-sets over a monoid S.

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37

Oldknow, Adrian. "On Twin Peaks and Flat Functions." Mathematical Gazette 79, no. 484 (1995): 47. http://dx.doi.org/10.2307/3619988.

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38

Kozłowski, Wojciech. "A remark on semi-∇-flat functions." Annales Polonici Mathematici 89, no. 2 (2006): 147–49. http://dx.doi.org/10.4064/ap89-2-4.

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Pan, Hong, Yonghong Jia, Dawei Zhao, Tianyu Xiu, and Fuzhi Duan. "A Tidal Flat Wetlands Delineation and Classification Method for High-Resolution Imagery." ISPRS International Journal of Geo-Information 10, no. 7 (2021): 451. http://dx.doi.org/10.3390/ijgi10070451.

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As an important part of coastal wetlands, tidal flat wetlands provide various significant ecological functions. Due to offshore pollution and unreasonable utilization, tidal flats have been increasingly threatened and degraded. Therefore, it is necessary to protect and restore this important wetland by monitoring its distribution. Considering the multiple sizes of research objects, remote sensing images with high resolutions have unique resolution advantages to support the extraction of tidal flat wetlands for subsequent monitoring. The purpose of this study is to propose and evaluate a tidal

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40

Shahraki, Ahmad Reza, Reza Abaee, and Elham Shahraki. "Omentum a Powerful Viable Organ in Patient with Liver Crash Trauma: A Case Series." EAS Journal of Medicine and Surgery 6, no. 02 (2024): 64–66. http://dx.doi.org/10.36349/easjms.2024.v06i02.010.

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The Omentum is a large flat adipose tissue layer nestling on the surface of the intra-peritoneal organs. Besides fat storage, omentum has key biological functions in immune-regulation and tissue regeneration. Omentum biological properties include neovascularization, haemostasis, tissue healing and regeneration and as an in vivo incubator for cells and tissue cultivation. Some of these properties have long been noted in surgical practice and used empirically in several procedures. we did 5 pedicular omental flap in 5 patients to prepare hemostasis in trauma with liver crash after resection and

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Anagnostopoulou, V., K. Díaz-Ordaz, O. Jenkinson, and C. Richard. "Entrance time functions for flat spot maps." Nonlinearity 23, no. 6 (2010): 1477–94. http://dx.doi.org/10.1088/0951-7715/23/6/011.

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PELED, RON, WOJCIECH SAMOTIJ, and AMIR YEHUDAYOFF. "Lipschitz Functions on Expanders are Typically Flat." Combinatorics, Probability and Computing 22, no. 4 (2013): 566–91. http://dx.doi.org/10.1017/s0963548313000163.

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This work studies the typical behaviour of random integer-valued Lipschitz functions on expander graphs with sufficiently good expansion. We consider two families of functions: M-Lipschitz functions (functions which change by at most M along edges) and integer-homomorphisms (functions which change by exactly 1 along edges). We prove that such functions typically exhibit very small fluctuations. For instance, we show that a uniformly chosen M-Lipschitz function takes only M+1 values on most of the graph, with a double exponential decay for the probability of taking other values.

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Gardner, Robert B., Marcus Kriele, and Udo Simon. "Generalized spherical functions on projectively flat manifolds." Results in Mathematics 27, no. 1-2 (1995): 41–50. http://dx.doi.org/10.1007/bf03322268.

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Peled, Ron. "High-dimensional Lipschitz functions are typically flat." Annals of Probability 45, no. 3 (2017): 1351–447. http://dx.doi.org/10.1214/16-aop1089.

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Rim, Kyung Soo. "Maximal Functions over Hypersurfaces with Flat Points." Journal of Mathematical Analysis and Applications 260, no. 1 (2001): 70–82. http://dx.doi.org/10.1006/jmaa.2001.7436.

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Wu, Gaofei, and Matthew Geoffrey Parker. "On Boolean functions with several flat spectra." Cryptography and Communications 11, no. 5 (2018): 853–80. http://dx.doi.org/10.1007/s12095-018-0309-2.

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Juhas, Anamarija, and Ladislav A. Novak. "Maximally Flat Waveforms with Finite Number of Harmonics in Class-FPower Amplifiers." Mathematical Problems in Engineering 2013 (2013): 1–9. http://dx.doi.org/10.1155/2013/169590.

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In this paper general solution to the problem of finding maximally flat waveforms with finite number of harmonics (maximally flat trigonometric polynomials) is provided. Waveform coefficients are expressed in closed form as functions of harmonic orders. Two special cases of maximally flat waveforms (so-called maximally flat even harmonic and maximally flat odd harmonic waveforms), which proved to play an important role in class-Fand inverse class-Fpower amplifier (PA) operations, are also considered. For these two special types of waveforms, coefficients are expressed as functions of two param

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Hochreiter, Sepp, and Jürgen Schmidhuber. "Flat Minima." Neural Computation 9, no. 1 (1997): 1–42. http://dx.doi.org/10.1162/neco.1997.9.1.1.

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We present a new algorithm for finding low-complexity neural networks with high generalization capability. The algorithm searches for a “flat” minimum of the error function. A flat minimum is a large connected region in weight space where the error remains approximately constant. An MDL-based, Bayesian argument suggests that flat minima correspond to “simple” networks and low expected overfitting. The argument is based on a Gibbs algorithm variant and a novel way of splitting generalization error into underfitting and overfitting error. Unlike many previous approaches, ours does not require ga

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Sazeedeh, Reza. "Gorenstein injective, Gorenstein flat modules and the section functor." Journal of Pure and Applied Algebra 211, no. 3 (2007): 773–83. http://dx.doi.org/10.1016/j.jpaa.2007.04.013.

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Lundström, Niklas L. P. "Estimates for -harmonic functions vanishing on a flat." Nonlinear Analysis: Theory, Methods & Applications 74, no. 18 (2011): 6852–60. http://dx.doi.org/10.1016/j.na.2011.06.041.

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