It has recently been shown that a suitably interconnected network of tunable telegraphic noise generators or "p-bits" can be used to perform even precise arithmetic functions like a 32-bit adder. In ...this letter, we use simulations based on the stochastic Landau-Lifshitz-Gilbert (sLLG) equation to demonstrate that similar impressive functions can be performed using unstable nanomagnets with energy barriers as low as a fraction of a kT. This is surprising because the magnetization of low-barrier nanomagnets is not telegraphic with discrete values of ±1. Rather, it fluctuates randomly among all values between -1 and +1, and the output magnets are read with a thresholding device that translates all positive values to one and all negative values to zero. We present sLLG-based simulations demonstrating the operation of a 32-bit adder, with a network of several hundred nanomagnets, exhibiting a remarkably precise correlation: The input magnets {A} and {B} as well as the output magnets {S} all fluctuate randomly and yet the quantity A + B-S is sharply peaked around zero! If we fix {A} and {B}, the sum magnets {S} rapidly converge to a unique state with S = A + B so that the system acts as an adder. But unlike standard adders, the operation is invertible. If we fix {S} and {B}, the remaining magnets {A} converge to the difference A = S - B. These examples emphasize a new direction for the field of nanomagnetics away from stable high-barrier magnets toward stochastic low-barrier magnets that not only operate with lower currents, but are also more promising for continued downscaling.
Zeros of recursively defined polynomials Heim, Bernhard; Neuhauser, Markus; Tröger, Robert
Journal of difference equations and applications,
04/2020, Letnik:
26, Številka:
4
Journal Article
Recenzirano
Let g, h be two arithmetic functions. In this paper, we study zero-free intervals of polynomials of the type
We use a difference equation approach. Let
be the divisor sum function and
the identity ...function, then the zeros of the polynomials dictate the vanishing properties of the coefficients of all powers of the Dedekind eta function. This is directly related to the Lehmer conjecture on the non-vanishing of the Ramanujan τ-function. Let
be the identity function. Then, the polynomials are related to generalized Laguerre polynomials for
and to Chebyshev polynomials of the second kind for
. The paper is furnished with several interesting examples. We found polynomials with decreasing minimal real zeros. We provide a simple arithmetic function g to approximate the minimal real zeros attached to σ.
For a wide class of sequences of integer domains
D
n
⊂
N
d
,
n
∈
N
, we prove distributional limit theorems for
F
(
X
1
(
n
)
,
…
,
X
d
(
n
)
)
, where
F
is a multivariate multiplicative function and
...(
X
1
(
n
)
,
…
,
X
d
(
n
)
)
is a random vector with uniform distribution on
D
n
. As a corollary, we obtain limit theorems for the greatest common divisor and least common multiple of the random set
{
X
1
(
n
)
,
…
,
X
d
(
n
)
}
. This generalizes previously known limit results for
D
n
being either a discrete cube or a discrete hyperbolic region.
We generalize certain recent results of Ushiroya concerning Ramanujan expansions of arithmetic functions of two variables. We also show that some properties on expansions of arithmetic functions of ...one and several variables using classical and unitary Ramanujan sums, respectively, run parallel.
Engineering DNA logic systems is considered as one of the most promising strategies for next-generation molecular computers. Owing to the inherent features of DNA, such as low cost, easy synthesis, ...and controllable hybridization, various DNA logic devices with different functions have been developed in the recent decade. Besides, a variety of logic-programmed biological applications are also explored, which initiates a new chapter for DNA logic computing. Although this field has gained rapid developments, a systematical review that could not only elaborate the logic principles of diverse DNA logic devices but also outline recent representative works is urgently needed. In this review, we first elaborate the general classification and logical principle of diverse DNA logic devices, in which the operating strategy of these devices and representative examples are selectively presented. Then, we review state-of-the-art advancements in DNA computing based on different non-canonical DNA-nanostructures during the past decade, in which some classical works are summarized. After that, the innovative applications of DNA computing to logic-controlled bioanalysis, cell imaging, and drug load/delivery are selectively presented. Finally, we analyze current obstacles and suggest appropriate prospects for this area.
We unify in a large class of additive functions the results obtained in the first part of this work. The proof rests on series involving the Riemann zeta function and certain sums of primes which may ...have their own interest.
The Mellin transform and several Dirichlet series related with the Riemann zeta function are used to deduce some identities similar to the classical Müntz formula 4. These formulas are derived in the ...critical strip and in the half-plane Re(s)<0. As particular cases, integral representations for products of the gamma and zeta functions are exhibited.
Let
N
be the set of positive integers, and denote by
λ
(
A
)
=
inf
{
t
>
0
:
∑
a
∈
A
a
-
t
<
∞
}
the convergence exponent of
A
⊂
N
. For
0
<
q
≤
1
,
0
≤
q
≤
1
, respectively, the admissible ideals
I
...<
q
,
I
≤
q
of all subsets
A
⊂
N
with
λ
(
A
)
<
q
,
λ
(
A
)
≤
q
, respectively, satisfy
I
<
q
⊊
I
c
(
q
)
⊊
I
≤
q
, where
I
c
(
q
)
=
{
A
⊂
N
:
∑
a
∈
A
a
-
q
<
∞
}
.
In this note we sharpen the results of Baláž et al. from (J Number Theory 183:74–83, 2018) and other papers, concerning characterizations of
I
c
(
q
)
-convergence of various arithmetic functions in terms of
q
. This is achieved by utilizing
I
<
q
- and
I
≤
q
-convergence, for which new methods and criteria are developed.
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