scholarly journals ALMOST ALL PRIMES HAVE A MULTIPLE OF SMALL HAMMING WEIGHT

2016 ◽  
Vol 94 (2) ◽  
pp. 224-235 ◽  
Author(s):  
CHRISTIAN ELSHOLTZ
Keyword(s):  
Log P ◽  
Hamming Weight ◽  
Multiplicative Subgroup ◽  
Image Position ◽  
Almost All

We improve recent results of Bourgain and Shparlinski to show that, for almost all primes $p$, there is a multiple $mp$ that can be written in binary as $$\begin{eqnarray}mp=1+2^{m_{1}}+\cdots +2^{m_{k}},\quad 1\leq m_{1}<\cdots <m_{k},\end{eqnarray}$$ with $k=6$ (corresponding to Hamming weight seven). We also prove that there are infinitely many primes $p$ with a multiplicative subgroup $A=\langle g\rangle \subset \mathbb{F}_{p}^{\ast }$, for some $g\in \{2,3,5\}$, of size $|A|\gg p/(\log p)^{3}$, where the sum–product set $A\cdot A+A\cdot A$ does not cover $\mathbb{F}_{p}$ completely.

DNA Interaction and Biological Activities of Heteroleptic Palladium (II) Complexes

2021 ◽  
Vol 43 (2) ◽  
pp. 227-227
Author(s):  
Muhammad Anwar Saeed Muhammad Anwar Saeed ◽  
Hizbullah Khan Hizbullah Khan ◽  
Muhammad Sirajuddin Muhammad Sirajuddin ◽  
Syed Muhammad Salman Syed Muhammad Salman
Keyword(s):  
Biological Activities ◽  
Thiobarbituric Acid ◽  
Dna Interaction ◽  
Log P ◽  
Bacterial Strains ◽  
Experimental Conditions ◽  
H2o2 Treatment ◽  
Biological Studies ◽  
Almost All ◽  
Anti Fungal

The manuscript describes the binding of DNA as well as biological studies of some mixed ligand dithiocarbamate Palladium (II) complexes (1-5). The observed compounds are of general formulae [PdCl(DT)(PR3)]. The dithiocarbamate “DT” and “PR3” groups are varied among the studied complexes as DT = bis[(2-methoxyethyl) dithiocarbamate)] (1 and 2), dibutyl dithiocarbamate (4 and 5), bis[(2-ethyl) hexyl dithiocarbamate)] (3); PR3 = triphenyl phosphine (1), benzy diphenyl phosphine (2), diphenyl-tert-butyl phpsphine (3), diphenyl-p-tolyl phosphine (4) and diphenyl-2-methoxy phenyl phosphine (5). The synthesized complexes were screened for DNA binding study via (UV Visible spectrophotometry and Viscometery) and biological activities such as anti-bacterial and anti-fungal, Molinspiration calculations and antioxidant potencies stimulated by hydrogen peroxide in human blood lymphocytes. In case of drug DNA interaction, complexes showed some sort of interaction with DNA solution. Almost all the complexes exhibited moderate antifungal and antibacterial behavior (against Gram positive and negative bacterial strains). The Molinspiration calculation study revealed that the said Pd (II) mixed complexes are biologically significant drugs having adequate molecular properties regarding drug likeness, except the log P values of complexes 3-5 because some structural adjustments must be done for enhancement of their bioavailability and hydrophilic nature. Regarding the antioxidant potential of complexes 1, 2 and 4, the H2O2 treatment of complexes violently decreased the action of antioxidant enzymes, superoxide dismutase and catalase and enhanced the level of thiobarbituric acid-reacting substances. Under experimental conditions, we conclude that all complexes act as anti-mutagens as they significantly suppress H2O2-induced oxidative damage at non-genotoxic concentrations.

On a Stein And Weiss Property of the Conjugate Function

1990 ◽  
Vol 42 (1) ◽  
pp. 109-125
Author(s):  
Nakhlé Asmar
Keyword(s):  
Fourier Transform ◽  
Abelian Group ◽  
Conjugate Function ◽  
Locally Compact Abelian Group ◽  
Locally Compact ◽  
Character Group ◽  
Locally Compact Abelian Groups ◽  
Compact Abelian Groups ◽  
Image Position ◽  
Almost All

(1.1) The conjugate function on locally compact abelian groups. Let G be a locally compact abelian group with character group Ĝ. Let μ denote a Haar measure on G such that μ(G) = 1 if G is compact. (Unless stated otherwise, all the measures referred to below are Haar measures on the underlying groups.) Suppose that Ĝ contains a measurable order P: P + P ⊆P; PU(-P)= Ĝ; and P⋂(—P) =﹛0﹜. For ƒ in ℒ2(G), the conjugate function of f (with respect to the order P) is the function whose Fourier transform satisfies the identity for almost all χ in Ĝ, where sgnP(χ)= 0, 1, or —1, according as χ =0, χ ∈ P\\﹛0﹜, or χ ∈ (—P)\﹛0﹜.

Small Solutions of the Congruence ax2 + by2 ≡ c(mod k)

1969 ◽  
Vol 12 (3) ◽  
pp. 311-320 ◽  
Author(s):  
Kenneth S. Williams
Keyword(s):  
Asymptotic Formula ◽  
Absolute Constant ◽  
Log P ◽  
Positive Absolute Constant ◽  
Image Position

In 1957, Mordell [3] provedTheorem. If p is an odd prime there exist non-negative integers x, y ≤ A p3/4 log p, where A is a positive absolute constant, such that(1.1)provided (abc, p) = 1.Recently Smith [5] has obtained a sharp asymptotic formula for the sum where r(n) denotes the number of representations of n as the sum of two squares.

Discounted branching random walks

1985 ◽  
Vol 17 (01) ◽  
pp. 53-66
Author(s):  
K. B. Athreya
Keyword(s):  
Functional Equation ◽  
Random Walks ◽  
Unique Solution ◽  
Log P ◽  
Branching Random Walks ◽  
Probabilistic Construction ◽  
Image Position

Let F(·) be a c.d.f. on [0,∞), f(s) = ∑∞ 0 pjsi a p.g.f. with p 0 = 0, &lt; 1 &lt; m = Σj p j &lt; ∞ and 1 &lt; ρ &lt;∞. For the functional equation for a c.d.f. H(·) on [0,∞] we establish that if 1 – F(x) = O(x –θ ) for some θ &gt; α =(log m)/(log p) there exists a unique solution H(·) to (∗) in the class C of c.d.f.’s satisfying 1 – H(x) = o(x –α ). We give a probabilistic construction of this solution via branching random walks with discounting. We also show non-uniqueness if the condition 1 – H(x) = o(x –α ) is relaxed.

A non-Markovian birth process with logarithmic growth

1975 ◽  
Vol 12 (04) ◽  
pp. 673-683
Author(s):  
G. R. Grimmett
Keyword(s):  
Random Variables ◽  
Independent Random Variables ◽  
Birth Process ◽  
Image Position ◽  
Almost All ◽  
Increasing Function ◽  
Logarithmic Growth

I show that the sumof independent random variables converges in distribution when suitably normalised, so long as theXksatisfy the following two conditions:μ(n)= E |Xn|is comparable withE|Sn| for largen,andXk/μ(k) converges in distribution. Also I consider the associated birth processX(t) = max{n:Sn≦t} when eachXkis positive, and I show that there exists a continuous increasing functionv(t) such thatfor some variableYwith specified distribution, and for almost allu. The functionv, satisfiesv(t) =A(1 +o(t)) logt. The Markovian birth process with parameters λn= λn, where 0 &lt; λ &lt; 1, is an example of such a process.

Numbers badly approximate by fractions with prime denominator

1995 ◽  
Vol 118 (1) ◽  
pp. 1-5 ◽  
Author(s):  
Glyn Harman
Keyword(s):  
Continued Fraction ◽  
Arithmetic Progressions ◽  
The Other ◽  
Strong Result ◽  
Infinitely Many Solutions ◽  
Continued Fraction Expansion ◽  
Primes In Arithmetic Progressions ◽  
Prime Variable ◽  
Image Position ◽  
Almost All

We write ‖x‖ to denote the least distance from x to an integer, and write p for a prime variable. Duffin and Schaeffer [l] showed that for almost all real α the inequalityhas infinitely many solutions if and only ifdiverges. Thus f(x) = (x log log (10x))−1 is a suitable choice to obtain infinitely many solutions for almost all α. It has been shown [2] that for all real irrational α there are infinitely many solutions to (1) with f(p) = p−/13. We will show elsewhere that the exponent can be increased to 7/22. A very strong result on primes in arithmetic progressions (far stronger than anything within reach at the present time) would lead to an improvement on this result. On the other hand, it is very easy to find irrational a such that no convergent to its continued fraction expansion has prime denominator (for example (45– √10)/186 does not even have a square-free denominator in its continued fraction expansion, since the denominators are alternately divisible by 4 and 9).

Projection theorems for box and packing dimensions

1996 ◽  
Vol 119 (2) ◽  
pp. 287-295 ◽  
Author(s):  
K. J. Falconer ◽  
J. D. Howroyd
Keyword(s):  
Orthogonal Projection ◽  
Packing Dimension ◽  
Analytic Subset ◽  
Box Counting ◽  
Packing Dimensions ◽  
Image Position ◽  
Almost All ◽  
Projection Theorems

AbstractWe show that if E is an analytic subset of ℝn thenfor almost all m–dimensional subspaces V of ℝn, where projvE is the orthogonal projection of E onto V and dimp denotes packing dimension. The same inequality holds for lower and upper box counting dimensions, and these inequalities are the best possible ones.

Independent dominating sets in graphs of girth five

2020 ◽  
pp. 1-16
Author(s):  
Ararat Harutyunyan ◽  
Paul Horn ◽  
Jacques Verstraete
Keyword(s):  
Minimum Degree ◽  
Dominating Set ◽  
Regularity Conditions ◽  
Dominating Sets ◽  
Vertex Graph ◽  
First Results ◽  
Complete Bipartite Graphs ◽  
Image Position ◽  
Almost All ◽  
Independent Dominating Set

Abstract Let $\gamma(G)$ and $${\gamma _ \circ }(G)$$ denote the sizes of a smallest dominating set and smallest independent dominating set in a graph G, respectively. One of the first results in probabilistic combinatorics is that if G is an n-vertex graph of minimum degree at least d, then $$\begin{equation}\gamma(G) \leq \frac{n}{d}(\log d + 1).\end{equation}$$ In this paper the main result is that if G is any n-vertex d-regular graph of girth at least five, then $$\begin{equation}\gamma_(G) \leq \frac{n}{d}(\log d + c)\end{equation}$$ for some constant c independent of d. This result is sharp in the sense that as $d \rightarrow \infty$ , almost all d-regular n-vertex graphs G of girth at least five have $$\begin{equation}\gamma_(G) \sim \frac{n}{d}\log d.\end{equation}$$ Furthermore, if G is a disjoint union of ${n}/{(2d)}$ complete bipartite graphs $K_{d,d}$ , then ${\gamma_\circ}(G) = \frac{n}{2}$ . We also prove that there are n-vertex graphs G of minimum degree d and whose maximum degree grows not much faster than d log d such that ${\gamma_\circ}(G) \sim {n}/{2}$ as $d \rightarrow \infty$ . Therefore both the girth and regularity conditions are required for the main result.

On a variant of sum-product estimates and explicit exponential sum bounds in prime fields

2009 ◽  
Vol 146 (1) ◽  
pp. 1-21 ◽  
Author(s):  
J. BOURGAIN ◽  
M. Z. GARAEV
Keyword(s):  
Prime Order ◽  
Exponential Sums ◽  
Exponential Sum ◽  
Complex Numbers ◽  
Number Of Solutions ◽  
Prime Fields ◽  
Explicit Bounds ◽  
Multiplicative Subgroup ◽  
Image Position

AbstractLet Fp be the field of a prime order p and F*p be its multiplicative subgroup. In this paper we obtain a variant of sum-product estimates which in particular implies the bound for any subset A ⊂ Fp with 1 < |A| < p12/23. Then we apply our estimate to obtain explicit bounds for some exponential sums in Fp. We show that for any subsets X, Y, Z ⊂ F*p and any complex numbers αx, βy, γz with |αx| ≤ 1, |βy| ≤ 1, |γz| ≤ 1, the following bound holds: We apply this bound further to show that if H is a subgroup of F*p with |H| > p1/4, then Finally we show that if g is a generator of F*p then for any M < p the number of solutions of the equation is less than $M^{3-1/24+o(1)}\Bigl(1+(M^2/p)^{1/24}\Bigr).$. This implies that if p1/2 < M < p, then

scholarly journals Marstrand-type Theorems for the Counting and Mass Dimensions in ℤ

2016 ◽  
Vol 25 (5) ◽  
pp. 700-743 ◽  
Author(s):  
DANIEL GLASSCOCK
Keyword(s):  
Recent Work ◽  
Orthogonal Projection ◽  
Side Length ◽  
Orthogonal Projections ◽  
Mass Dimension ◽  
Image Position ◽  
Almost All

The counting and (upper) mass dimensions of a set A ⊆ $\mathbb{R}^d$ are $$D(A) = \limsup_{\|C\| \to \infty} \frac{\log | \lfloor A \rfloor \cap C |}{\log \|C\|}, \quad \smash{\overline{D}}\vphantom{D}(A) = \limsup_{\ell \to \infty} \frac{\log | \lfloor A \rfloor \cap [-\ell,\ell)^d |}{\log (2 \ell)},$$ where ⌊A⌋ denotes the set of elements of A rounded down in each coordinate and where the limit supremum in the counting dimension is taken over cubes C ⊆ $\mathbb{R}^d$ with side length ‖C‖ → ∞. We give a characterization of the counting dimension via coverings: $$D(A) = \text{inf} \{ \alpha \geq 0 \mid {d_{H}^{\alpha}}(A) = 0 \},$$ where $${d_{H}^{\alpha}}(A) = \lim_{r \rightarrow 0} \limsup_{\|C\| \rightarrow \infty} \inf \biggl\{ \sum_i \biggl(\frac{\|C_i\|}{\|C\|} \biggr)^\alpha \ \bigg| \ 1 \leq \|C_i\| \leq r \|C\| \biggr\}$$ in which the infimum is taken over cubic coverings {Ci} of A ∩ C. Then we prove Marstrand-type theorems for both dimensions. For example, almost all images of A ⊆ $\mathbb{R}^d$ under orthogonal projections with range of dimension k have counting dimension at least min(k, D(A)); if we assume D(A) = D(A), then the mass dimension of A under the typical orthogonal projection is equal to min(k, D(A)). This work extends recent work of Y. Lima and C. G. Moreira.

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