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Whereas the design and properties of bent and plateaued functions have been frequently addressed during the past few decades, there are only a few design methods of so-called 5-valued spectra Boolean functions whose Walsh spectra takes the…

Whether two distinct APN functions can have a Hamming distance of $1$ remains an open problem. In 2020, L. Budaghyan et al. introduced a new CCZ-invariant $\Pi_F$ which can be used to provide lower bounds on the Hamming distance between a…

We show that there is a simple (approximately radial) function on $\reals^d$, expressible by a small 3-layer feedforward neural networks, which cannot be approximated by any 2-layer network, to more than a certain constant accuracy, unless…

In a prior paper [14], along with P. Ellingsen, P. Felke and A. Tkachenko, we defined a new (output) multiplicative differential, and the corresponding c-differential uniformity, which has the potential of extending differential…

We prove a necessary condition for some polynomials of degree 4e (e an odd number) to be APN over F q n for large n, and we investigate the polynomials f of degree 12.

A Walsh zero space (WZ space) for $f:F_{2^n}\rightarrow F_{2^n}$ is an $n$-dimensional vector subspace of $F_{2^n}\times F_{2^n}$ whose all nonzero elements are Walsh zeros of $f$. We provide several theoretical and computer-free…

The set of linear structures of most known balanced Boolean functions is nontrivial. In this paper, some balanced Boolean functions whose set of linear structures is trivial are constructed. We show that any APN function in even dimension…

The vectorial nonlinearity of a vector valued function is its distance from the set of affine functions. In 2017, Liu, Mesnager and Chen conjectured a general upper bound for the vectorial linearity. Recently, Carlet proved a lower bound in…

We address the problem of which functions can arise as Dehn functions of K\"ahler groups. We explain why there are examples of K\"ahler groups with linear, quadratic, and exponential Dehn function. We then proceed to show that there is an…

In a previous article the authors determined the best-known upper bound for the cardinality of the image set for several classes of functions, including planar functions. Here, we show that the upper bound cannot be tight for planar…

We contribute to a better understanding of the class of functions that can be represented by a neural network with ReLU activations and a given architecture. Using techniques from mixed-integer optimization, polyhedral theory, and tropical…

The augmented cube $AQ_n$, proposed by Choudum and Sunitha [S. A. Choudum, V. Sunitha, Augmented cubes, Networks 40 (2) (2002) 71-84], is a $(2n-1)$-regular $(2n-1)$-connected graph $(n\ge 4)$. This paper determines that the 2-extra…

The Gauss hypergeometric functions 2F1 with arbitrary values of parameters are reduced to two functions with fixed values of parameters, which differ from the original ones by integers. It is shown that in the case of integer and/or…

Given a finite set of real numbers $A$, the generalised golden ratio is the unique real number $\mathcal{G}(A) > 1$ for which we only have trivial unique expansions in smaller bases, and have non-trivial unique expansions in larger bases.…

We consider an infinite family of exponents $e(l,k)$ with two parameters, $l$ and $k$, and derive sufficient conditions for $e(l,k)$ to be 0-APN over $\mathbb{F}_{2^n}$. These conditions allow us to generate, for each choice of $l$ and $k$,…

Sidorov and Vershik showed that in base $G=\frac{\sqrt{5}+1}{2}$ and with the digits $0,1$ the numbers $x=nG ~(\text {mod} 1)$ have $\aleph_{0}$ expansions for any $n\in\mathbb{Z}$, while the other elements of $(0, \frac{1}{G-1})$ have…

In 2021, Calderini et al. introduced a construction for APN functions on $\mathbb{F}_{2^{2m}}$ in bivariate form $$ f(x,y)=\big(xy,\, x^{2^r+1} + x^{2^{r+m/2}} y^{2^{m/2}} + bxy^{2^r} + cy^{2^r+1}\big),\quad r < m/2,\quad \gcd(r, m) = 1. $$…

We study the universality of complex-valued neural networks with bounded widths and arbitrary depths. Under mild assumptions, we give a full description of those activation functions $\varrho:\mathbb{C}\to \mathbb{C}$ that have the property…

It is known that crooked functions can be used to construct many interesting combinatorial objects, and a quadratic function is crooked if and only if it is almost perfect nonlinear (APN). In this paper, we introduce two infinite classes of…

We find the sharp range for boundedness of the discrete bilinear spherical maximal function for dimensions $d \geq 5$. That is, we show that this operator is bounded on $l^{p}(\mathbb{Z}^d)\times l^{q}(\mathbb{Z}^d) \to l^{r}(\mathbb{Z}^d)$…