On Inversion in Z_{2^n-1}

In this paper we determined explicitly the multiplicative inverses of the Dobbertin and Welch APN exponents in Z_{2^n-1}, and we described the binary weights of the inverses of the Gold and Kasami exponents. We studied the function \de(n), which for a fixed positive integer d maps integers n\geq 1 to the least positive residue of the inverse of d modulo 2^n-1, if it exists. In particular, we showed that the function \de is completely determined by its values for 1 \leq n \leq \ordb, where \ordb is the order of 2 modulo the largest odd divisor of d.Comment: The first part of this work is an extended version of the results presented in ISIT1

Engaging Ancient Islamic Traditions in the Poetry of Saleha Ghabesh

This paper explores the integration of ancient Islamic heritage in Emirati literature, particularly the history of the rise and fall of the Muslim Empire in Andalusia, in an attempt to confront regional challenges and international transformations in the current era. Navigating the intersection between heritage and identity, the Emirati poet, Saleha Ghabesh, attempts to incorporate the ancient Islamic heritage in Andalusia as a dynamics of liberation in order to articulate domestic issues integral to the geopolitics of the United Arab Emirates and the Arab region in the age of globalization. Transforming the mythic history of Andalusia into a narrative of disclosure, the poet encounters a web of traditions and policies responsible for significant ramifications in the UAE and the Arab world. In a related context, the paper points out that the technique of adaptation, used by Ghabesh, which includes recollection rephrasing and re-writing of ancient heritage and Andalusian legacies to fulfill contemporary purposes, is part of the issue of hybridity and interculturation, characterizing the contemporary experience of political and cultural globalization. By assimilating heritage and historical traditions into contemporary Emirati literature, Ghabesh aims to link the past with the present reconstructing ancient narratives which shaped the cultural mythology of the Arab people

Crooked maps in F2n

AbstractA map f:F2n→F2n is called crooked if the set {f(x+a)+f(x):x∈F2n} is an affine hyperplane for every fixed a∈F2n∗ (where F2n is considered as a vector space over F2). We prove that the only crooked power maps are the quadratic maps x2i+2j with gcd(n,i−j)=1. This is a consequence of the following result of independent interest: for any prime p and almost all exponents 0⩽d⩽pn−2 the set {xd+γ(x+a)d:x∈Fpn} contains n linearly independent elements, where γ and a≠0 are arbitrary elements from Fpn

Metabolic Dependencies in Pancreatic Cancer.

Pancreatic ductal adenocarcinoma (PDA) is a highly lethal cancer with a long-term survival rate under 10%. Available cytotoxic chemotherapies have significant side effects, and only marginal therapeutic efficacy. FDA approved drugs currently used against PDA target DNA metabolism and DNA integrity. However, alternative metabolic targets beyond DNA may prove to be much more effective. PDA cells are forced to live within a particularly severe microenvironment characterized by relative hypovascularity, hypoxia, and nutrient deprivation. Thus, PDA cells must possess biochemical flexibility in order to adapt to austere conditions. A better understanding of the metabolic dependencies required by PDA to survive and thrive within a harsh metabolic milieu could reveal specific metabolic vulnerabilities. These molecular requirements can then be targeted therapeutically, and would likely be associated with a clinically significant therapeutic window since the normal tissue is so well-perfused with an abundant nutrient supply. Recent work has uncovered a number of promising therapeutic targets in the metabolic domain, and clinicians are already translating some of these discoveries to the clinic. In this review, we highlight mitochondria metabolism, non-canonical nutrient acquisition pathways (macropinocytosis and use of pancreatic stellate cell-derived alanine), and redox homeostasis as compelling therapeutic opportunities in the metabolic domain

Constructing irreducible polynomials recursively with a reverse composition method

We suggest a construction of the minimal polynomial $m_{\beta^k}$ of $\beta^k\in \mathbb F_{q^n}$ over $\mathbb F_q$ from the minimal polynomial $f= m_\beta$ for all positive integers $k$ whose prime factors divide $q-1$. The computations of our construction are carried out in $\mathbb F_q$. The key observation leading to our construction is that for $k \mid q-1$ holds $$m_{\beta^k}(X^k) = \prod_{j=1}^{\frac kt} \zeta_k^{-jn} f (\zeta_k^j X),$$ where $t= \max \{m\mid \gcd(n,k): f (X) = g (X^m), g \in \mathbb F_q[X]\}$ and $\zeta_{k}$ is a primitive $k$-th root of unity in $\mathbb F_q$. The construction allows to construct a large number of irreducible polynomials over $\mathbb F_q$ of the same degree. Since different applications require different properties, this large number allows the selection of the candidates with the desired properties

Physical, Chemical, and biotic influences on Zooplankton Composition in Zaranik Lagoon, Egypt

EnZaranik Protected Area encompasses the eastern end of Lake Bardawil: the Zaranik Lagoon. The lagoon is shallow, with numerous small islets scattered throughout it, most of which are covered with dense saltmarsh vegetation. Nitrogenous and phosphorus forms (ammonia, nitrite, nitrate, orthophosphate and total phosphorus) were studied as a basic nutrient salts affected different flora and fauna of the studied area. Nitrite was depleted completely during the study period except for winter. The nitrate values were fluctuated in a relatively narrow range (23.5 – 60 µg/l). Ammonia was detected in a normal range varied between 89-172 µg/l. Both orthophosphate and total phosphorus exhibit similar distribution dynamics. A total of 45 zooplankton species belonging to 9 main groups (Protista, Copepoda, Rotifera, Cladocera, Pteropoda, Cheatognatha, Cnidaria, Appendiculariae, and meroplankton) were recorded. Copepoda were the most abundant and ubiquitous zooplankton organisms in Zaranik protectorate, forming the 63 % of total zooplankton density. Salinity showed a negative correlation with total Protista (r = - 0.77) while NH3 showed a positive correlation with total zooplankton (r = 0.68)

Image sets of perfectly nonlinear maps

We present a lower bound on the image size of a $d$-uniform map, $d\geq 1$, of finite fields, by extending the methods used for planar maps. In the particularly interesting case of APN maps on binary fields, our bound coincides with the one obtained by Ingo Czerwinski, using a linear programming method. We study properties of APN maps of $\mathbb{F}_{2^n}$ with minimal image set. In particular, we observe that for even $n$, a Dembowski-Ostrom polynomial of form $f(x) =f'(x^3)$ is APN if and only if $f$ is almost-3-to-1, that is when its image set is minimal. We show that any almost-3-to-1 quadratic map is APN, if $n$ is even. For $n$ odd, we present APN Dembowski-Ostrom polynomials on $\mathbb{F}_{2^n}$ with image sizes $2^{n-1}$ and $5\cdot 2^{n-3}$. We present several results connecting the image sets of special APN maps with their Walsh spectrum. Especially, we show that a large class of APN maps has the classical Walsh spectrum. Finally, we prove that the image size of a non-bijective almost bent map contains at most $2^n-2^{(n-1)/2}$ elements.Comment: Minor revision with new references; Theorems 18, 19 are adde