In this paper, we prove that the weighted BMO space
BMO
p
(
ω
)
=
{
f
∈
L
loc
1
:
sup
Q
‖
χ
Q
‖
L
p
(
ω
)
−
1
‖
(
f
−
f
Q
)
ω
−
1
χ
Q
‖
L
p
(
ω
)
<
∞
}
is independent of the scale
p
∈ (0, ∞) in sense ...of norm when
ω
∈
A
1
. Moreover, we can replace
L
p
(
ω
) by
L
p
,∞
(
ω
). As an application, we characterize this space by the boundedness of the bilinear commutators
b, T
j
(
j
= 1, 2), generated by the bilinear convolution type Calderón-Zygmund operators and the symbol
b
, from
L
p
1
(
ω
)
×
L
p
2
(
ω
)
to
L
p
(
ω
1−
p
) with 1 <
p
1
,
p
2
< ∞ and 1/
p
=1/
p
1
+ 1/
p
2
. Thus we answer the open problem proposed by Chaffee affirmatively.
In this article, we analyze the boundedness for the fractional bilinear Hardy operators on variable exponent weighted Morrey–Herz spaces
M
K
˙
q
,
p
(
⋅
)
α
(
⋅
)
,
λ
(
w
)
. Similar estimates are ...obtained for their commutators when the symbol functions belong to
BMO
space with variable exponents.
We find a minimal notion of non-degeneracy for bilinear singular integral operators
T
and identify testing conditions on the multiplying function
b
that characterize the
L
p
×
L
q
→
L
r
,
1
<
p
,
q
<
...∞
and
r
>
1
2
,
boundedness of the bilinear commutator
b
,
T
1
(
f
,
g
) =
b
T
(
f
,
g
) −
T
(
b
f
,
g
). Our arguments cover almost all arrangements of the integrability exponents
p
,
q
,
r
with a single open problem presented in the end. Additionally, the arguments extend to the multilinear setting.
Lattice-based cryptography (LBC) has emerged as the most viable substitutes to the classical cryptographic schemes as 5 out of 7 finalist schemes in the 3rd round of the NIST post-quantum ...cryptography (PQC) standardization process are lattice based in construction. This work explores novel architectural optimizations in the FPGA-based hardware implementation of polynomial multiplication, which is a bottleneck in every LBC construction. To target ultra-high throughput, both schoolbook polynomial multiplication (SPM) and number theoretic transform (NTT) are explored: a completely parallel architecture of an SPM is undertaken while for NTT, radix-2 and radix-<inline-formula><tex-math notation="LaTeX">2^2</tex-math> <mml:math><mml:msup><mml:mn>2</mml:mn><mml:mn>2</mml:mn></mml:msup></mml:math><inline-graphic xlink:href="liu-ieq1-3144101.gif"/> </inline-formula> multi-path delay commutator (MDC) based pipelined architectures are adopted. Our proposed high-speed SPM (HSPM) structure on latest Xilinx UltraScale+ FPGA is 5× faster than the state-of-the-art LBC designs. Whereas, the proposed high-speed NTT (HNTT) structure (i.e., R<inline-formula><tex-math notation="LaTeX">2^2</tex-math> <mml:math><mml:msup><mml:mn>2</mml:mn><mml:mn>2</mml:mn></mml:msup></mml:math><inline-graphic xlink:href="liu-ieq2-3144101.gif"/> </inline-formula>MDC) takes only 0.63<inline-formula><tex-math notation="LaTeX">\mu</tex-math> <mml:math><mml:mi>μ</mml:mi></mml:math><inline-graphic xlink:href="liu-ieq3-3144101.gif"/> </inline-formula>s for the encryption, hence achieving the highest throughput of 408 Mbps. Moreover, all of the proposed designs achieve highest design efficiencies (i.e., throughput per slice (TPS)) in comparison to available LBC designs.
In this paper, pseudo-differential operators with homogeneous symbol classes associated with the Weinstein transform are introduced. The boundedness of pseudo-differential operators and commutator ...between two pseudo-differential operators on
ℌ
α
,2
r
are proven with the help of the Weinstein transform technique.
In this article, we study the commutators of Hausdorff operators and establish their boundedness on the weighted Herz spaces in the setting of the Heisenberg group.
This work is devoted to describing of constructions and main characteristics of some types compact pulse generators with amplitude of voltage pulses up to 200 kV, rise-times about 10 ns and pulse ...energy per pulse up to 20 J. For reduction of rise-time of high-voltage pulses were used spark-discharge two electrodes commutators (sharpeners) at working pressures up to 100 atm.
Probabilistic nilpotence in infinite groups Martino, Armando; Tointon, Matthew C. H.; Valiunas, Motiejus ...
Israel journal of mathematics,
09/2021, Letnik:
244, Številka:
2
Journal Article
Recenzirano
Odprti dostop
The ‘degree of
k
-step nilpotence’ of a finite group
G
is the proportion of the tuples (
x
1
,…,
x
k
+1
∈
G
k
+1
for which the simple commutator
x
1
, …,
x
k
+1
is equal to the identity. In this ...paper we study versions of this for an infinite group
G
, with the degree of nilpotence defined by sampling
G
in various natural ways, such as with a random walk, or with a Følner sequence if
G
is amenable. In our first main result we show that if
G
is finitely generated, then the degree of
k
-step nilpotence is positive if and only if
G
is virtually
k
-step nilpotent (Theorem 1.5). This generalises both an earlier result of the second author treating the case
k
= 1 and a result of Shalev for finite groups, and uses techniques from both of these earlier results. We also show, using the notion of polynomial mappings of groups developed by Leibman and others, that to a large extent the degree of nilpotence does not depend on the method of sampling (Theorem 1.12). As part of our argument we generalise a result of Leibman by showing that if
ϕ
is a polynomial mapping into a torsion-free nilpotent group, then the set of roots of
ϕ
is sparse in a certain sense (Theorem 5.1). In our second main result we consider the case where
G
is residually finite but not necessarily finitely generated. Here we show that if the degree of
k
-step nilpotence of the finite quotients of
G
is uniformly bounded from below, then
G
is virtually
k
-step nilpotent (Theorem 1.19), answering a question of Shalev. As part of our proof we show that degree of nilpotence of finite groups is sub-multiplicative with respect to quotients (Theorem 1.21), generalising a result of Gallagher.
We introduce intermediate commutators and study their degrees. We define
(
q
,
{
}
)
-capable groups and prove that a group
G
is
(
q
,
{
}
)
-capable if and only if
Z
(
q
,
{
}
)
∧
(
G
)
=
1
.
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