On Standard Derived Equivalences of Orbit Categories

Let k be a commutative ring, \(\mathcal {A}\) and \(\mathcal {B}\) – two k-linear categories with an action of a group G. We introduce the notion of a standard G-equivalence from \(\mathcal {K}_{p}^{\mathrm {b}}\mathcal {B}\) to \(\mathcal {K}_{p}^{\mathrm {b}}\mathcal {A}\), where \(\mathcal {K}_{p}^{\mathrm {b}}\mathcal {A}\) is the homotopy category of finitely generated projective \(\mathcal {A}\)-complexes. We construct a map from the set of standard G-equivalences to the set of standard equivalences from \(\mathcal {K}_{p}^{\mathrm {b}}\mathcal {B}\) to \(\mathcal {K}_{p}^{\mathrm {b}}\mathcal {A}\) and a map from the set of standard G-equivalences from \(\mathcal {K}_{p}^{\mathrm {b}}\mathcal {B}\) to \(\mathcal {K}_{p}^{\mathrm {b}}\mathcal {A}\) to the set of standard equivalences from \(\mathcal {K}_{p}^{\mathrm {b}}(\mathcal {B}/G)\) to \(\mathcal {K}_{p}^{\mathrm {b}}(\mathcal {A}/G)\), where \(\mathcal {A}/G\) denotes the orbit category. We investigate the properties of these maps and apply our results to the case where \(\mathcal {A}=\mathcal {B}=R\) is a Frobenius k-algebra and G is the cyclic group generated by its Nakayama automorphism ν. We apply this technique to obtain the generating set of the derived Picard group of a Frobenius Nakayama algebra over an algebraically closed field.     

Double ramification cycles on the moduli spaces of curves

Curves of genus \(g\) which admit a map to \(\mathbf {P}^{1}\) with specified ramification profile \(\mu\) over \(0\in \mathbf {P}^{1}\) and \(\nu\) over \(\infty\in \mathbf {P}^{1}\) define a double ramification cycle \(\mathsf{DR}_{g}(\mu,\nu)\) on the moduli space of curves. The study of the restrictions of these cycles to the moduli of nonsingular curves is a classical topic. In 2003, Hain calculated the cycles for curves of compact type. We study here double ramification cycles on the moduli space of Deligne-Mumford stable curves.The cycle \(\mathsf{DR}_{g}(\mu,\nu)\) for stable curves is defined via the virtual fundamental class of the moduli of stable maps to rubber. Our main result is the proof of an explicit formula for \(\mathsf{DR}_{g}(\mu,\nu)\) in the tautological ring conjectured by Pixton in 2014. The formula expresses the double ramification cycle as a sum over stable graphs (corresponding to strata classes) with summand equal to a product over markings and edges. The result answers a question of Eliashberg from 2001 and specializes to Hain’s formula in the compact type case.When \(\mu=\nu=\emptyset\), the formula for double ramification cycles expresses the top Chern class \(\lambda_{g}\) of the Hodge bundle of \(\overline {\mathcal{M}}_{g}\) as a push-forward of tautological classes supported on the divisor of non-separating nodes. Applications to Hodge integral calculations are given.     

Ladders and canonical varieties of completely regular semigroups

A completely regular semigroup is a (disjoint) union of its (maximal) subgroups. We consider it here with the unary operation of inversion within its maximal subgroups. Their totality \(\mathcal {C}\mathcal {R}\) forms a variety whose lattice of subvarieties is denoted by \(\mathcal {L}(\mathcal {C}\mathcal {R})\). On it, one defines the relations \(\mathbf {B}^\wedge \) and \(\mathbf {B}^\vee \) by

$$\begin{aligned} \begin{array}{lll} \mathcal {U}\ \mathbf {B}^\wedge \ \mathcal {V}&{} \Longleftrightarrow &{} \mathcal {U}\cap \mathcal {B} =\mathcal {V}\cap \mathcal {B}, \\ \mathcal {U}\ \mathbf {B}^\vee \ \mathcal {V}&{} \Longleftrightarrow &{} \mathcal {U}\vee \mathcal {B} =\mathcal {V}\vee \mathcal {B} , \end{array} \end{aligned}$$

respectively, where \(\mathcal {B}\) denotes the variety of all bands. This is a study of the interplay between the \(\cap \)-subsemilatice \(\triangle \) of \(\mathcal {L}(\mathcal {C}\mathcal {R})\) of upper ends of \(\mathbf {B}^\wedge \)-classes and their \(\mathbf {B}^\vee \)-classes. The main tool is the concept of a ladder and their \(\mathbf {B}^\vee \)-classes, an indispensable part of the important Polák’s theorem providing a construction for the join of varieties of completely regular semigroups. The paper includes the tables of ladders of the upper ends of most \(\mathbf {B}^\wedge \)-classes. Canonical varieties consist of two ascending countably infinite chains which generate most of the upper ends of \(\mathbf {B}^\wedge \)-classes.

     

Fractional Sobolev metrics on spaces of immersed curves

Motivated by applications in the field of shape analysis, we study reparametrization invariant, fractional order Sobolev-type metrics on the space of smooth regular curves \(\mathrm {Imm}(\mathrm {S}^{1},\mathbb {R}^d)\) and on its Sobolev completions \({\mathcal {I}}^{q}(\mathrm {S}^{1},{\mathbb {R}}^{d})\). We prove local well-posedness of the geodesic equations both on the Banach manifold \({\mathcal {I}}^{q}(\mathrm {S}^{1},{\mathbb {R}}^{d})\) and on the Fréchet-manifold \(\mathrm {Imm}(\mathrm {S}^{1},\mathbb {R}^d)\) provided the order of the metric is greater or equal to one. In addition we show that the \(H^s\)-metric induces a strong Riemannian metric on the Banach manifold \({\mathcal {I}}^{s}(\mathrm {S}^{1},{\mathbb {R}}^{d})\) of the same order s, provided \(s>\frac{3}{2}\). These investigations can be also interpreted as a generalization of the analysis for right invariant metrics on the diffeomorphism group.     

Duality of Non-Homogeneous Central Herz-Morrey-Musielak-Orlicz Spaces

We determine the associate space of non-homogeneous central Herz-Morrey-Musielak-Orlicz space \(\mathcal {H}^{\Phi ,q,\omega }(\textbf {R}^{N})\). We also determine the associate spaces of the space \(\underline {\mathcal {H}}^{\Phi ,q,\omega }(\textbf {R}^{N})\) and its complementary space \(\overline {\mathcal {H}}^{\Phi ,q,\omega }(\textbf {R}^{N})\).     

Approximation of Invariant Subspaces in Some Dirichlet-Type Spaces

The Hilbert space \(\mathcal {D}_{2}\) is the space of all holomorphic functions f defined on the open unit disc \(\mathbb {D}\) such that \({f}^{'}\) is in the Hardy Hilbert space \(\mathbf {H}^2.\) In this paper, we prove that the invariant subspaces of \(\mathcal {D}_{2}\) with respect to multiplication operator \(M_{z}\) can be approximated with finite co-dimensional invariant subspaces. We also obtain a partial result in this direction for the classical Dirichlet space.     

Teichmüller curves in genus three and just likely intersections in $\mathbf{G}_{m}^{n}\times\mathbf{G}_{a}^{n}$

We prove that the moduli space of compact genus three Riemann surfaces contains only finitely many algebraically primitive Teichmüller curves. For the stratum \(\Omega\mathcal{M}_{3}(4)\), consisting of holomorphic one-forms with a single zero, our approach to finiteness uses the Harder-Narasimhan filtration of the Hodge bundle over a Teichmüller curve to obtain new information on the locations of the zeros of eigenforms. By passing to the boundary of moduli space, this gives explicit constraints on the cusps of Teichmüller curves in terms of cross-ratios of six points on \(\mathbf{P}^{1}\).These constraints are akin to those that appear in Zilber and Pink’s conjectures on unlikely intersections in diophantine geometry. However, in our case one is lead naturally to the intersection of a surface with a family of codimension two algebraic subgroups of \(\mathbf{G}_{m}^{n}\times\mathbf{G}_{a}^{n}\) (rather than the more standard \(\mathbf{G}_{m}^{n}\)). The ambient algebraic group lies outside the scope of Zilber’s Conjecture but we are nonetheless able to prove a sufficiently strong height bound.For the generic stratum \(\Omega\mathcal{M}_{3}(1,1,1,1)\), we obtain global torsion order bounds through a computer search for subtori of a codimension-two subvariety of \(\mathbf{G}_{m}^{9}\). These torsion bounds together with new bounds for the moduli of horizontal cylinders in terms of torsion orders yields finiteness in this stratum. The intermediate strata are handled with a mix of these techniques.     

Largest initial segments pointwise fixed by automorphisms of models of set theory

Given a model \(\mathcal {M}\) of set theory, and a nontrivial automorphism j of \(\mathcal {M}\), let \(\mathcal {I}_{\mathrm {fix}}(j)\) be the submodel of \(\mathcal {M}\) whose universe consists of elements m of \(\mathcal {M}\) such that \(j(x)=x\) for every x in the transitive closure of m (where the transitive closure of m is computed within \(\mathcal {M}\)). Here we study the class \(\mathcal {C}\) of structures of the form \(\mathcal {I}_{\mathrm {fix}}(j)\), where the ambient model \(\mathcal {M}\) satisfies a frugal yet robust fragment of \(\mathrm {ZFC}\) known as \(\mathrm {MOST}\), and \(j(m)=m\) whenever m is a finite ordinal in the sense of \(\mathcal {M}.\) Our main achievement is the calculation of the theory of \(\mathcal {C}\) as precisely \(\mathrm {MOST+\Delta }_{0}^{\mathcal {P}}\)-\(\mathrm {Collection}\). The following theorems encapsulate our principal results: Theorem A. Every structure in \(\mathcal {C}\) satisfies \(\mathrm {MOST+\Delta }_{0}^{\mathcal {P}}\)-\(\mathrm { Collection}\). Theorem B. Each of the following three conditions is sufficient for a countable structure \(\mathcal {N}\) to be in \(\mathcal {C}\):(a) \(\mathcal {N}\) is a transitive model of \(\mathrm {MOST+\Delta }_{0}^{\mathcal {P}}\)-\(\mathrm {Collection}\).(b) \(\mathcal {N}\) is a recursively saturated model of \(\mathrm {MOST+\Delta }_{0}^{\mathcal {P}}\)-\(\mathrm {Collection}\).(c) \(\mathcal {N}\) is a model of \(\mathrm {ZFC}\). Theorem C. Suppose \(\mathcal {M}\) is a countable recursively saturated model of \(\mathrm {ZFC}\) and I is a proper initial segment of \(\mathrm {Ord}^{\mathcal {M}}\) that is closed under exponentiation and contains \(\omega ^\mathcal {M}\) . There is a group embedding \(j\longmapsto \check{j}\) from \(\mathrm {Aut}(\mathbb {Q})\) into \(\mathrm {Aut}(\mathcal {M})\) such that I is the longest initial segment of \(\mathrm {Ord}^{\mathcal {M}}\) that is pointwise fixed by \(\check{j}\) for every nontrivial \(j\in \mathrm {Aut}(\mathbb {Q}).\) In Theorem C, \(\mathrm {Aut}(X)\) is the group of automorphisms of the structure X, and \(\mathbb {Q}\) is the ordered set of rationals.     

The Generalized Entropy Ergodic Theorem for Nonhomogeneous Markov Chains

Let \((\xi _n)_{n=0}^\infty \) be a nonhomogeneous Markov chain taking values in a finite state-space \(\mathbf {X}=\{1,2,\ldots ,b\}\). In this paper, we will study the generalized entropy ergodic theorem with almost-everywhere and \(\mathcal {L}_1\) convergence for nonhomogeneous Markov chains; this generalizes the corresponding classical results for Markov chains.     

Positive Systems of Kostant Roots

Let \(\mathfrak {g}\) be a simple complex Lie algebra and let \(\mathfrak {t} \subset \mathfrak {g}\) be a toral subalgebra of \(\mathfrak {g}\). As a \(\mathfrak {t}\)-module \(\mathfrak {g}\) decomposes as

$$\mathfrak{g} = \mathfrak{s} \oplus \left( \oplus_{\nu \in \mathcal{R}}~ \mathfrak{g}^{\nu}\right)$$

where \(\mathfrak {s} \subset \mathfrak {g}\) is the reductive part of a parabolic subalgebra of \(\mathfrak {g}\) and \(\mathcal {R}\) is the Kostant root system associated to \(\mathfrak {t}\). When \(\mathfrak {t}\) is a Cartan subalgebra of \(\mathfrak {g}\) the decomposition above is nothing but the root decomposition of \(\mathfrak {g}\) with respect to \(\mathfrak {t}\); in general the properties of \(\mathcal {R}\) resemble the properties of usual root systems. In this note we study the following problem: “Given a subset \(\mathcal {S} \subset \mathcal {R}\), is there a parabolic subalgebra \(\mathfrak {p}\) of \(\mathfrak {g}\) containing \(\mathcal {M} = \oplus _{\nu \in \mathcal {S}} \mathfrak {g}^{\nu }\) and whose reductive part equals \(\mathfrak {s}\)?”. Our main results is that, for a classical simple Lie algebra \(\mathfrak {g}\) and a saturated \(\mathcal {S} \subset \mathcal {R}\), the condition \((\text {Sym}^{\cdot }(\mathcal {M}))^{\mathfrak {s}} = \mathbb {C}\) is necessary and sufficient for the existence of such a \(\mathfrak {p}\). In contrast, we show that this statement is no longer true for the exceptional Lie algebras F₄,E₆,E₇, and E₈. Finally, we discuss the problem in the case when \(\mathcal {S}\) is not saturated.

     

Sparse Recovery from Inaccurate Saturated Measurements

This article studies a variation of the standard compressive sensing problem, in which sparse vectors \(\mathbf{x}\in\mathbb{R}^{N}\) are acquired through inaccurate saturated measurements \(\mathbf{y}= \mathcal{S}(\mathbf {A}\mathbf{x}+ \mathbf{e}) \in\mathbb{R}^{m}\), \(m \ll N\). The saturation function \(\mathcal{S}\) acts componentwise by sending entries that are large in absolute value to plus-or-minus a threshold while keeping the other entries unchanged. The present study focuses on the effect of the presaturation error \(\mathbf{e}\in\mathbb{R}^{m}\). The existing theory for accurate saturated measurements, i.e., the case \(\mathbf{e}= \mathbf{0}\), which exhibits two regimes depending on the magnitude of \(\mathbf{x}\in\mathbb {R}^{N}\), is extended here. A recovery procedure based on convex optimization is proposed and shown to be robust to presaturation error in both regimes. Another procedure ignoring the presaturation error is also analyzed and shown to be robust in the small magnitude regime.     

Sharp MSE Bounds for Proximal Denoising

Denoising has to do with estimating a signal \(\mathbf {x}_0\) from its noisy observations \(\mathbf {y}=\mathbf {x}_0+\mathbf {z}\). In this paper, we focus on the “structured denoising problem,” where the signal \(\mathbf {x}_0\) possesses a certain structure and \(\mathbf {z}\) has independent normally distributed entries with mean zero and variance \(\sigma ^2\). We employ a structure-inducing convex function \(f(\cdot )\) and solve \(\min _\mathbf {x}\{\frac{1}{2}\Vert \mathbf {y}-\mathbf {x}\Vert _2^2+\sigma {\lambda }f(\mathbf {x})\}\) to estimate \(\mathbf {x}_0\), for some \(\lambda >0\). Common choices for \(f(\cdot )\) include the \(\ell _1\) norm for sparse vectors, the \(\ell _1-\ell _2\) norm for block-sparse signals and the nuclear norm for low-rank matrices. The metric we use to evaluate the performance of an estimate \(\mathbf {x}^*\) is the normalized mean-squared error \(\text {NMSE}(\sigma )=\frac{{\mathbb {E}}\Vert \mathbf {x}^*-\mathbf {x}_0\Vert _2^2}{\sigma ^2}\). We show that NMSE is maximized as \(\sigma \rightarrow 0\) and we find the exact worst-case NMSE, which has a simple geometric interpretation: the mean-squared distance of a standard normal vector to the \({\lambda }\)-scaled subdifferential \({\lambda }\partial f(\mathbf {x}_0)\). When \({\lambda }\) is optimally tuned to minimize the worst-case NMSE, our results can be related to the constrained denoising problem \(\min _{f(\mathbf {x})\le f(\mathbf {x}_0)}\{\Vert \mathbf {y}-\mathbf {x}\Vert _2\}\). The paper also connects these results to the generalized LASSO problem, in which one solves \(\min _{f(\mathbf {x})\le f(\mathbf {x}_0)}\{\Vert \mathbf {y}-{\mathbf {A}}\mathbf {x}\Vert _2\}\) to estimate \(\mathbf {x}_0\) from noisy linear observations \(\mathbf {y}={\mathbf {A}}\mathbf {x}_0+\mathbf {z}\). We show that certain properties of the LASSO problem are closely related to the denoising problem. In particular, we characterize the normalized LASSO cost and show that it exhibits a “phase transition” as a function of number of observations. We also provide an order-optimal bound for the LASSO error in terms of the mean-squared distance. Our results are significant in two ways. First, we find a simple formula for the performance of a general convex estimator. Secondly, we establish a connection between the denoising and linear inverse problems.     

Maps with the Unique Extension Property and $$\hbox {C}^{*}$$-Extreme Points

We investigate boundary representations in the context where Hilbert spaces are replaced by \(\hbox {C}^{*}\)-modules over abelian von Neumann algebras and apply this to study \(\hbox {C}^{*}\)-extreme points. We present an (unexpected) example of a weak* compact \(\mathcal {B}\)-convex subset of \({\mathbb {B}}(\mathcal {H})\) without \(\mathcal {B}\)-extreme points, where \(\mathcal {B}\) is an abelian von Neumann algebra on a Hilbert space \(\mathcal {H}\). On the other hand, if \(\mathcal {A}\) is a von Neumann algebra with a separable predual and whose finite part is injective, we show that each weak* compact \(\mathcal {A}\)-convex subset of \(\ell ^{\infty }(\mathcal {A})\) is generated by its \(\mathcal {A}\)-extreme points.     

Absolute continuity and singularity of Palm measures of the Ginibre point process

We prove a dichotomy between absolute continuity and singularity of the Ginibre point process \(\mathsf {G}\) and its reduced Palm measures \(\{\mathsf {G}_{\mathbf {x}}, \mathbf {x} \in \mathbb {C}^{\ell }, \ell = 0,1,2\ldots \}\), namely, reduced Palm measures \(\mathsf {G}_{\mathbf {x}}\) and \(\mathsf {G}_{\mathbf {y}}\) for \(\mathbf {x} \in \mathbb {C}^{\ell }\) and \(\mathbf {y} \in \mathbb {C}^{n}\) are mutually absolutely continuous if and only if \(\ell = n\); they are singular each other if and only if \(\ell \not = n\). Furthermore, we give an explicit expression of the Radon–Nikodym density \(d\mathsf {G}_{\mathbf {x}}/d \mathsf {G}_{\mathbf {y}}\) for \(\mathbf {x}, \mathbf {y} \in \mathbb {C}^{\ell }\).     

A deterministic sparse FFT algorithm for vectors with small support

In this paper we consider the special case where a signal x\({\in }\,\mathbb {C}^{N}\) is known to vanish outside a support interval of length m < N. If the support length m of x or a good bound of it is a-priori known we derive a sublinear deterministic algorithm to compute x from its discrete Fourier transform \(\widehat {\mathbf x}\,{\in }\,\mathbb {C}^{N}\). In case of exact Fourier measurements we require only \({\mathcal O}\)(m\(\log \)m) arithmetical operations. For noisy measurements, we propose a stable \({\mathcal O}\)(m\(\log \)N) algorithm.     

On counting subring-submodules of free modules over finite commutative frobenius rings

Let \(\texttt {R}\) be a finite commutative Frobenius ring and \(\texttt {S}\) a Galois extension of \(\texttt {R}\) of degree m. For positive integers k and \(k'\), we determine the number of free \(\texttt {S}\)-submodules \(\mathcal {B}\) of \(\texttt {S}^\ell \) with the property \(k=\texttt {rank}_\texttt {S}(\mathcal {B})\) and \(k'=\texttt {rank}_\texttt {R}(\mathcal {B}\cap \texttt {R}^\ell )\). This corrects the wrong result (Bill in Linear Algebr Appl 22:223–233, 1978, Theorem 6) which was given in the language of codes over finite fields.     

Theorems on Large Deviations for Randomly Indexed Sum of Weighted Random Variables

In this paper, we consider a random variable \(Z_{t}=\sum_{i=1}^{N_{t}}a_{i}X_{i}\), where \(X, X_{1}, X_{2}, \ldots\) are independent identically distributed random variables with mean E X=μ and variance D X=σ ²>0. It is assumed that Z ₀=0, 0≤a _i <∞, and N _t , t≥0 is a non-negative integer-valued random variable independent of X _i , i=1,2,…?. The paper is devoted to the analysis of accuracy of the standard normal approximation to the sum \(\tilde{Z}_{t}=(\mathbf{D}Z_{t})^{-1/2}(Z_{t}-\mathbf{E}Z_{t})\), large deviation theorems in the Cramer and power Linnik zones, and exponential inequalities for \(\mathbf{P}(\tilde{Z}_{t}\geq x)\).     

Consistency of the intensional level of the Minimalist Foundation with Church’s thesis and axiom of choice

Consistency with the formal Church’s thesis, for short CT, and the axiom of choice, for short AC, was one of the requirements asked to be satisfied by the intensional level of a two-level foundation for constructive mathematics as proposed by Maietti and Sambin (in Crosilla, Schuster (eds) From sets and types to topology and analysis: practicable foundations for constructive mathematics, Oxford University Press, Oxford, 2005). Here we show that this is the case for the intensional level of the two-level Minimalist Foundation, for short MF, completed in 2009 by the second author. The intensional level of MF consists of an intensional type theory à la Martin-Löf, called mTT. The consistency of mTT with CT and AC is obtained by showing the consistency with the formal Church’s thesis of a fragment of intensional Martin-Löf’s type theory, called \(\mathbf{MLtt}_1\), where mTT can be easily interpreted. Then to show the consistency of \(\mathbf{MLtt}_1\) with CT we interpret it within Feferman’s predicative theory of non-iterative fixpoints \(\widehat{ID_1}\) by extending the well known Kleene’s realizability semantics of intuitionistic arithmetics so that CT is trivially validated. More in detail the fragment \(\mathbf{MLtt}_1\) we interpret consists of first order intensional Martin-Löf’s type theory with one universe and with explicit substitution rules in place of usual equality rules preserving type constructors (hence without the so called \(\xi \)-rule which is not valid in our realizability semantics). A key difficulty encountered in our interpretation was to use the right interpretation of lambda abstraction in the applicative structure of natural numbers in order to model all the equality rules of \(\mathbf{MLtt}_1\) correctly. In particular the universe of \(\mathbf{MLtt}_1\) is modelled by means of \(\widehat{ID_1}\)-fixpoints following a technique due first to Aczel and used by Feferman and Beeson.     

Idempotent and <Emphasis Type="Italic">p</Emphasis>-potent quadratic functions: distribution of nonlinearity and co-dimension

The Walsh transform \(\widehat{Q}\) of a quadratic function \(Q:{\mathbb F}_{p^n}\rightarrow {\mathbb F}_p\) satisfies \(|\widehat{Q}(b)| \in \{0,p^{\frac{n+s}{2}}\}\) for all \(b\in {\mathbb F}_{p^n}\), where \(0\le s\le n-1\) is an integer depending on Q. In this article, we study the following three classes of quadratic functions of wide interest. The class \(\mathcal {C}_1\) is defined for arbitrary n as \(\mathcal {C}_1 = \{Q(x) = \mathrm{Tr_n}(\sum _{i=1}^{\lfloor (n-1)/2\rfloor }a_ix^{2^i+1})\;:\; a_i \in {\mathbb F}_2\}\), and the larger class \(\mathcal {C}_2\) is defined for even n as \(\mathcal {C}_2 = \{Q(x) = \mathrm{Tr_n}(\sum _{i=1}^{(n/2)-1}a_ix^{2^i+1}) + \mathrm{Tr_{n/2}}(a_{n/2}x^{2^{n/2}+1}) \;:\; a_i \in {\mathbb F}_2\}\). For an odd prime p, the subclass \(\mathcal {D}\) of all p-ary quadratic functions is defined as \(\mathcal {D} = \{Q(x) = \mathrm{Tr_n}(\sum _{i=0}^{\lfloor n/2\rfloor }a_ix^{p^i+1})\;:\; a_i \in {\mathbb F}_p\}\). We determine the generating function for the distribution of the parameter s for \(\mathcal {C}_1, \mathcal {C}_2\) and \(\mathcal {D}\). As a consequence we completely describe the distribution of the nonlinearity for the rotation symmetric quadratic Boolean functions, and in the case \(p > 2\), the distribution of the co-dimension for the rotation symmetric quadratic p-ary functions, which have been attracting considerable attention recently. Our results also facilitate obtaining closed formulas for the number of such quadratic functions with prescribed s for small values of s, and hence extend earlier results on this topic. We also present the complete weight distribution of the subcodes of the second order Reed–Muller codes corresponding to \(\mathcal {C}_1\) and \(\mathcal {C}_2\) in terms of a generating function.