-
Article
On some determinants involving the tangent function
Let p be an odd prime and let \(a,b\in {\mathbb {Z}}\) a , ...
-
Article
Open AccessDeficiency of ASGR1 promotes liver injury by increasing GP73-mediated hepatic endoplasmic reticulum stress
Liver injury is a core pathological process in the majority of liver diseases, yet the genetic factors predisposing individuals to its initiation and progression remain poorly understood. Here we show that asi...
-
Article
On Certain Determinants and Related Legendre Symbols
Let p be an odd prime. For \(b,c\in {\mathbb {Z}}\) b , ...
-
Article
The tangent function and power residues modulo primes
Let p be an odd prime, and let a be an integer not divisible by p. When m is a positive integer with p ≡ 1 (mod 2m) and 2 is an mth power residue modulo p, we determine the value of the product
-
Article
Some parametric congruences involving generalized central trinomial coefficients
For \(n=0,1,2,\ldots \) n = 0 ...
-
Article
Representing \({\varvec{n}}\) as \({\varvec{n=x+y+z}}\) with \({\varvec{x^2+y^2+z^2}}\) a square
In this paper, we mainly confirm the following conjecture of Sun posed in 2013: Each positive integer n can be written as \(n=x+y+z\) ...
-
Article
Legendre Symbols Related to Certain Determinants
Let p be an odd prime. For \(b,c\in {\mathbb {Z}}\) b , ...
-
Article
Therapeutic Effects of Different Animal Bile Powders on Lipid Metabolism Disorders and Their Composition Analysis
To compare the therapeutic effect of different animal bile powders on lipid metabolism disorders induced by high-fat diet in rats, and analyze the bioactive components of each animal bile powder.
-
Article
On Sums and Products in a Field
We study sums and products in a field. Let F be a field with ch(F) ≠ 2, where ch(F) is the characteristic of F. For any integer k ⩾ 4, we show that any x ∈ F can be written as a1 + … + ak with a1, …, ak ∈ F and a
-
Article
Supercongruences involving Lucas sequences
For \(A,B\in {\mathbb {Z}}\) A , B ...
-
Article
On permutations of \(\{1,\ldots ,n\}\) and related topics
In this paper, we study combinatorial aspects of permutations of \(\{1,\ldots ,n\}\) {
-
Article
q-Analogues of Some Series for Powers of \(\pi \)
We obtain q-analogues of several series for powers of \(\pi \) π . For examp...
-
Article
Open AccessUreteral polyps protruding from the urethra
Ureteral fibro-epithelial polyp (UFP) is a rare benign ureteral tumor, and surgical removal of the polyps is still the preferred solution. Although many cases have reported polyps extending to the bladder, our...
-
Article
Further results on Hilbert’s Tenth Problem
Hilbert’s Tenth Problem (HTP) asked for an algorithm to test whether an arbitrary polynomial Diophantine equation with integer coefficients has solutions over the ring ℤ of integers. This was finally solved by...
-
Article
Universal sums of three quadratic polynomials
Let a, b, c, d, e and f be integers with a ⩾ c ⩾ e > 0, b > –a and b ≡ a (mod 2), d > –c and d ≡ c (mod 2), f > –e and f ≡ e (mod 2). Suppose that b ⩾ d if a = c, and d ⩾ f if c = e. When b(a–b), d(c–d) and f(e–f
-
Article
New congruences involving products of two binomial coefficients
Let \(p>3\) p > ...
-
Article
Nanostructured polyelectrolyte-surfactant complex pervaporation membranes for ethanol recovery: the relationship between the membrane structure and separation performance
Polyelectrolyte-surfactant complexes (PESCs) were fabricated through the interaction of poly(acrylic acid) and four different cationic surfactants or their mixtures. PESC membranes were prepared by solution ca...
-
Article
A new theorem on the prime-counting function
For \(x>0\) x > ...
-
Chapter and Conference Paper
Conjectures on Representations Involving Primes
We pose 100 new conjectures on representations involving primes or related things, which might interest number theorists and stimulate further research. Below are five typical examples: (i) For any positive in...
-
Article
A Novel Aβ B-Cell Epitope Vaccine (rCV01) for Alzheimer’s Disease Improved Synaptic and Cognitive Functions in 3 × Tg-AD Mice
Alzheimer’s disease (AD) is a neurodegenerative disorder characterized by a progressive amyloid-β accumulation, loss of cognitive abilities, and synaptic alterations. Given the remarkable recovery of cognition...
