We consider Presburger arithmetic (PA) extended by scalar multiplication an algebraic irrational number $\alpha$, and call this extension $\alpha$-Presburger ($\alpha$-PA). show that the complexity of deciding sentences in $\alpha$-PA is substantially harder than PA. Indeed, when $\alpha$ quadratic $r\geq 4$, with $r$ alternating quantifier blocks at most $c\ r$ variables inequalities requires ...

For a given bounded positive (semidefinite) linear operator $A$ on complex Hilbert space $\big(\mathcal{H}, \langle \cdot, \cdot\rangle \big)$, we consider the semi-Hilbertian \cdot\rangle_A \big)$ where ${\langle x, y\rangle}_A := Ax, y\rangle$ for every $x, y\in\mathcal{H}$. The $A$-numerical radius of an $A$-bounded $T$ $\mathcal{H}$ is by\[\omega_A(T)=\sup\Big\{\big|{\langle Tx, x\rangle}_A...

The lll algorithm is a polynomial-time for reducing d-dimensional lattice with exponential approximation factor. Currently, the most efficient variant of lll, by Neumaier and Stehlé, has theoretical running time in $$d^4\cdot B^{1+o\left( 1\right) }$$ where B bitlength entries, but never been implemented. This work introduces new asymptotically fast, parallel, yet heuristic, reduction algorithm...

Consider the random subgraph process on a base graph $G$ $n$ vertices: sequence $\lbrace G_t \rbrace _{t=0} ^{|E(G)|}$ of subgraphs obtained by choosing an ordering edges uniformly at random, and sequentially adding to $G_0$, empty vertex set $G$, according chosen ordering. We show that if has one following properties: 1. There is positive constant $\varepsilon > 0$ such $\delta (G) \geq \lef...

We )nd that adding certain amounts of inhibitory inputs to a neuron improves its capability of accurately decoding the input information. The optimal ratio r of inhibitory to excitatory inputs for decoding the input information from an observation of the e,erent interspike intervals is calculated. Surprisingly, the Fisher information could be zero for certain values of the ratio, seemingly impl...

We prove the following result: Theorem. Every algebraic distributive lattice D with at most א1 compact elements is isomorphic to the ideal lattice of a von Neumann regular ring R. (By earlier results of the author, the א1 bound is optimal.) Therefore, D is also isomorphic to the congruence lattice of a sectionally complemented modular lattice L, namely, the principal right ideal lattice of R. F...

We construct an Identity-Based Key Encapsulation Mechanism (IBKEM) in a generic “leveled” multilinear map setting and prove its security under multilinear decisional Diffie-Hellmanin assumption in the selective-ID model. Then, we make our IB-KEM translated to the GGH framework, which defined an “approximate” version of a multilinear group family from ideal lattices, and modify our proof of secu...

We study well-rounded lattices which come from ideals in quadratic number fields, generalizing some recent results of the first author with K. Petersen [8]. In particular, we give a characterization of ideal well-rounded lattices in the plane and show that a positive proportion of real and imaginary quadratic number fields contains ideals giving rise to well-rounded lattices.

If A, B are bounded linear operators on a complex Hilbert space, then we prove that $$\begin{aligned} w(A)\le & {} \frac{1}{2}\left( \Vert A\Vert +\sqrt{r\left( |A||A^*|\right) }\right) ,\\ w(AB \pm BA)\le 2\sqrt{2}\Vert B\Vert \sqrt{ w^2(A)-\frac{c^2(\mathfrak {R}(A))+c^2(\mathfrak {I}(A))}{2} }, \end{aligned}$$ where $$w(\cdot ),\left\| \cdot \right\| $$ , and $$r(\cdot )$$ the numerical radi...

A C *-algebra A is called an ideal C * -algebra (or equally a dual algebra) if it is an ideal in its bidual A**. M.C.F. Berglund proved that subalgebras and quotients of ideal C*-algebras are also ideal C*-algebras, that a commutative C *-algebra A is an ideal C *-algebra if and only if it is isomorphicto C (Q) for some discrete space ?. We investigate ideal J*-algebras and show that the a...