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The main focus of this article is to study the characteristics of charged pions and their correlation with shower particles emitted in the interaction of 84Kr + emulsion at 1 A GeV. Author have also studied the dependence of the average multiplicity of charged pions and shower particles with projectile mass. In this study author observed that the average multiplicity of charged pions and shower particles showing the linear dependency on the projectile mass.

This article concerns the multiplicity of solutions for a class of nonlocal and nonhomogeneous Neumann boundary value problems involving the p ( x )-Laplacian, in which both nonlinear terms assume critical growth. We use variational method, exploring an important truncation argument and properties of the genus.

Let Ω be a C 2 bounded domain in R N ( N ≥ 3 ), δ ( x ) = dist ( x , ∂ Ω ) and C H ( Ω ) be the best constant in the Hardy inequality with respect to Ω. We investigate positive solutions to a boundary value problem for Lane-Emden equations with Hardy potential of the form − Δ u − μ δ 2 u = u p in Ω , u = ρ ν on ∂ Ω , ( P ρ ) where 0 μ C H ( Ω ) , ρ is a positive parameter, ν is a positive Radon measure on ∂Ω with norm 1 and 1 p N μ , with N μ being a critical exponent depending on N and μ. It is known from [22] that there exists a threshold value ρ ⁎ such that problem ( P ρ ) admits a positive solution if 0 ρ ≤ ρ ⁎ , and no positive solution if ρ > ρ ⁎ . In this paper, we go further in the study of the solution set of ( P ρ ) . We show that the problem admits at least two positive solutions if 0 ρ ρ ⁎ and a unique positive solution if ρ = ρ ⁎ . We also prove the existence of at least two positive solutions for Lane-Emden systems { − Δ u − μ δ 2 u = v p in Ω , − Δ v − μ δ 2 v = u q in Ω , u = ρ ν , v = σ τ on ∂ Ω , under the smallness condition on the positive parameters ρ and σ.

Inspired by the work of Amitsur [1] on finite groups whose irreducible characters all have degree (multiplicity) 1 or 2, in this paper we study association schemes whose irreducible characters all have multiplicity 1 or 2. We will first show that the general case can be reduced to commutative association schemes. Then for commutative association schemes with multiplicities 1 or 2, we prove that their Krein parameters are all rational integers. Using automorphism groups of association schemes, we obtain a characterization and classification of those commutative association schemes all valencies and multiplicities of which are 1 or 2 in terms of Cayley schemes.

For the semilinear Dirichlet problem Δ u + g ( u ) = f ( x ) for x ∈ Ω , u = 0 on ∂ Ω decompose f ( x ) = μ 1 φ 1 + e ( x ) , where φ 1 is the principal eigenfunction of the Laplacian with zero boundary conditions, and e ( x ) ⊥ φ 1 in L 2 ( Ω ) , and similarly write u ( x ) = ξ 1 φ i + U ( x ) , with U ⊥ φ 1 in L 2 ( Ω ) . We study properties of the solution curve ( u ( x ) , μ 1 ) ( ξ 1 ) , and in particular its section μ 1 = μ 1 ( ξ 1 ) , which governs the multiplicity of solutions. We consider both general nonlinearities, and some important classes of equations, and obtain detailed description of solution curves under the assumption g ′ ( u ) λ 2 . We obtain particularly detailed results in case of one dimension. This approach is well suited for numerical computations, which we perform to illustrate our results.

In this article, periodic solutions from a fine focus $ U = 0 $, are accomplished for several classes. Some classes have polynomial coefficients, while the remaining classes $ C_{14, 7} $, $ C_{16, 8} $ and $ C_{5, 5}, $ $ C_{6, 6} $ have non-homogeneous and homogenous trigonometric coefficients accordingly. By adopting a systematic procedure of bifurcation that occurs under perturbation of the coefficients, we have succeeded to find the highest known multiplicity $ 10 $ as an upper bound for the class $ C_{9, 4} $, $ C_{11, 3} $ with algebraic and $ C_{5, 5}, $ $ C_{6, 6} $ with trigonometric coefficients. Polynomials of different degrees with various coefficients have been discussed using symbolic computation in Maple 18. All of the results are executed and validated by using past and present theory, and they were found to be novel and authentic in their respective domains.

We study the unitary principal series of the split group S p i n m ( F ) , where F is a p-adic field. Let χ ˜ be a unitary character of a maximal F-split torus T ˜ of G S p i n m ( F ) , and let χ be its restriction to T = T ˜ ∩ S p i n m ( F ) . The R-groups R χ ˜ and R χ of the corresponding principal series representations fit in the exact sequence 0 → R χ ˜ → R χ → R χ / R χ ˜ → 0 . We give a complete answer to the question of splitting of this exact sequence. We also prove that the multiplicity is one when irreducible constituents in unitary principal series are restricted from G S p i n ( F ) to S p i n ( F ) . Further, based on Keys' result, we prove Arthur's conjecture on R-groups for unitary principal series of S p i n .

In this paper we consider existence, nonexistence and multiplicity of solutions for a class of indefinite quasilinear elliptic problems in the upper half-space involving weights in anisotropic Lebesgue spaces. One of our basic tools consists in a Hardy type inequality proved in the present paper that allows us to establish Sobolev embeddings into Lebesgue spaces with weights in anisotropic Lebesgue spaces.

In this paper, using the variational principle, the existence and multiplicity of solutions for p x , q x -Kirchhoff type problem with Navier boundary conditions are proved. At the same time, the sufficient conditions for the multiplicity of solutions are obtained.

International audience; In this paper we consider the-stabilization of th-order linear time-invariant (LTI) dynamical systems using Multiplicity-Induced-Dominancy (MID)-based controller design in the presence of delays in the input or the output channels. A sufficient condition is given for the dominancy of a real root with multiplicity at least n + 1 and at least using an integral factorization of the corresponding characteristic function. A necessary condition for-stabilizability is analyzed utilizing the property that the derivative of \alpha-stable quasipolynomial is also-stable under certain conditions. Sufficient and necessary conditions are given for systems with real-rooted open loop characteristic function: the delay intervals are determined where the conditions for dominancy and \alpha-stabilizability are satisfied. The efficiency of the proposed controller design is shown in the case of a multi-link inverted pendulum.

In the paper, we study the existence of weak solutions for a class of new nonlocal problems involving a p x -Laplacian operator. By using Ekeland’s variational principle and mountain pass theorem, we prove that the new p x -Kirchhoff problem has at least two nontrivial weak solutions.

This paper deals with the existence, uniqueness and the multiplicity for the boundary value problem of a class of fractional differential equations involving the Riemann-Liouville fractional derivative with nonlocal integral boundary conditions. By using the properties of the Green's function, the Banach contraction principle, the xed point index theory and the Leggett-Williams fixed point theorem, one shows the existence and the uniqueness of positive solutions and the existence and multiplicity of positive solutions. As applications, some examples are presented at the end to illustrate the main results.

Particles 4(2), 275 - 287 (2021). doi:10.3390/particles4020024, The size and evolution of the matter created in relativistic heavy-ion collisions strongly depend on collision geometry, defined by the impact parameter. However, the impact parameter cannot be measured directly in an experiment but might be inferred from final state observables using the centrality procedure. We present the procedure of centrality determination for the Multi-Purpose Detector (MPD) at the NICA collider and its performance using the multiplicity of produced charged particles at midrapidity. The validity of the procedure is assessed using the simulated data for Au + Au collisions at $√s_{NN}$ = 4–11 GeV., Published by MDPI AG, Basel

We study existence of solutions for the fractional problem \begin{equation*} (P_m) \quad \left \{ \begin{aligned} (-��)^{s} u + ��u &=g(u) & \; \text{in $\mathbb{R}^N$}, \cr \int_{\mathbb{R}^N} u^2 dx &= m, & \cr u \in H^s_r&(\mathbb{R}^N), & \end{aligned} \right. \label{problemx} \end{equation*} where $N\geq 2$, $s\in (0,1)$, $m>0$, $��$ is an unknown Lagrange multiplier and $g \in C(\mathbb{R}, \mathbb{R})$ satisfies Berestycki-Lions type conditions. Using a Lagrange formulation of the problem $(P_m)$, we prove the existence of a weak solution with prescribed mass when $g$ has $L^2$ subcritical growth. The approach relies on the construction of a minimax structure, by means of a Pohozaev's mountain in a product space and some deformation arguments under a new version of the Palais-Smale condition introduced in [21,25]. A multiplicity result of infinitely many normalized solutions is also obtained if $g$ is odd., To be published in Nonlinearity (accepted)

In this paper, we study the following nonlinear magnetic Kirchhoff equation: { - ( a ⁢ ϵ 2 + b ⁢ ϵ ⁢ [ u ] A / ϵ 2 ) ⁢ Δ A / ϵ ⁢ u + V ⁢ ( x ) ⁢ u = f ⁢ ( | u | 2 ) ⁢ u in ⁢ ℝ 3 , u ∈ H 1 ⁢ ( ℝ 3 , ℂ ) , \left\{\begin{aligned} &\displaystyle{-}(a\epsilon^{2}+b\epsilon[u]_{A/% \epsilon}^{2})\Delta_{A/\epsilon}u+V(x)u=f(\lvert u\rvert^{2})u&&\displaystyle% \phantom{}\text{in }\mathbb{R}^{3},\\ &\displaystyle u\in H^{1}(\mathbb{R}^{3},\mathbb{C}),\end{aligned}\right. where ϵ > 0 {\epsilon>0} , a , b > 0 {a,b>0} are constants, V : ℝ 3 → ℝ {V:\mathbb{R}^{3}\rightarrow\mathbb{R}} and A : ℝ 3 → ℝ 3 {A:\mathbb{R}^{3}\rightarrow\mathbb{R}^{3}} are continuous potentials, and Δ A ⁢ u {\Delta_{A}u} is the magnetic Laplace operator. Under a local assumption on the potential V, by combining variational methods, a penalization technique and the Ljusternik–Schnirelmann theory, we prove multiplicity properties of solutions and concentration phenomena for ϵ small. In this problem, the function f is only continuous, which allows to consider larger classes of nonlinearities in the reaction.

In continuation of the work in Leventides and Petroulakis (Adv Appl Clifford Algebras 27:1503–1515, 2016), Leventides et al. (J Optim Theory Appl 169(1):1–16, 2016), which defines extremal varieties in $$\mathbb {P}\left( {{ \bigwedge ^2}{\mathbb {R}^n}} \right) $$ , we define a more general concept of extremal varieties of the real Grassmannian $${G_p}\left( {{\mathbb {R}^n}} \right) $$ in $$\mathbb {P}\left( {{ \bigwedge ^p}{\mathbb {R}^n}} \right) $$ . This concept is based on the minimization of the sums of squares of the quadratic Plucker relations defining the Grassmannian variety as well as the reverse maximisation problem. Such extremal problems define a set of Grassmannian inequalities on the set of Grassmann matrices, which are essential for the definition of the Grassmann variety and its dual extremal variety. We define and prove these inequalities for a general Grassmannian and we apply the existing results, in the cases $${{ \wedge ^2}{\mathbb {R}^{2n}}}$$ and $${{ \wedge ^n}{\mathbb {R}^{2n}}}$$ . The resulting extremal varieties underline the fact which was demonstrated in Leventides et al. (2016, Linear Algebra Appl 461:139–162, 2014), that such varieties are represented by multi-vectors that acquire the property of a unique singular value with total multiplicity. Crucial to these inequalities are the numbers $$M_{n,p}$$ , which are calculated within the cases mentioned above.

This paper concerns the critical points and the level sets of solutions of the Grushin equation in the plane. After exactly establishing descriptions about the critical points of the homogeneous Gruhin-harmonic polynomials and investigating the local geometric properties of the level sets near these critical points, we prove that the critical points of solutions of the Grushin equation are isolated and each critical point has finite multiplicity. We further estimate the numbers of interior critical points of solutions of the Dirichlet boundary value problem.

Let G be a graph (which may have multiple edges but no loops), let the multiplicity μ G ( u , v ) between two different vertices u and v be the number of edges joining u and v , and let μ G ( u ) = max { μ G ( u , v ) : v ∈ V ( G ) ∖ { u } } . In this paper, we prove that if G is triangle-free in which no two quadrangles share edges, then for any two integral functions a , b : V ( G ) ⟶ N ∖ { 0 , 1 } with d G ( v ) ≥ a ( v ) + b ( v ) + 2 μ G ( v ) − 3 for each vertex v of G , there is a partition ( A , B ) of V ( G ) such that d G [ A ] ( u ) ≥ a ( u ) for each u ∈ A and d G [ B ] ( v ) ≥ b ( v ) for each v ∈ B . Consequently, we confirm two conjectures of Schweser and Stiebitz (2019).

This paper provides the first Liapunov coefficient formula of 3D Lotka-Volterra equations. This formula gives applications to stability of positive equilibrium and to detecting sub/super criticality of Hopf bifurcation. For 3D competitive Lotka-Volterra equations, combining this formula with the Poincare-Bendixson theorem, we obtain criteria on multiplicity of limit cycles among Zeeman's classes 27-31, and present a series of examples to admit at least two limit cycles, which are rigorously proved by the first Liapunov coefficient formula, rather than by symbolic computation using Maple. A new Hopf bifurcation that all 2 × 2 principal minors of the community matrix are positive is found, and numerical simulation reveals its global limit cycle bifurcations are plenty.

We consider the problem of finding a condition for a univariate polynomial having a given multiplicity structure when the number of distinct roots is given. It is well known that such conditions can be written as conjunctions of several polynomial equations and one inequation in the coefficients, by using repeated parametric gcd's. In this paper, we give a novel condition which is not based on repeated gcd's. Furthermore, it is shown that the number of polynomials in the condition is optimal and the degree of polynomials is smaller than that in the previous condition based on repeated gcd's.

In this work, we study the existence and multiplicity of positive solutions for a nonlocal problem with changing sign data.

We investigate the multiplicity of solutions for the one-dimensional elliptic Dirichlet boundary value problem with jumping nonlinearities. We obtain three theorems depending on the source terms wh...

Let $$\mathcal {P}_r$$ denote an almost-prime with at most r prime factors, counted according to multiplicity. In this paper, we establish a theorem of Bombieri–Vinogradov type for the Piatetski–Shapiro primes $$p=[n^{1/\gamma }]$$ with $$\frac{85}{86}

The nullity of a graph, denoted by η ( G ) , is the multiplicity of zero as an eigenvalue of the adjacency matrix of G. Fiorini, Gutman and Sciriha (2005) in [5] presented an upper bound for the nullities of trees in terms of the order n and the maximum degree Δ. In our work we show that, under the same conditions, all possible nullities below that bound are attained. This result allows us to obtain an upper bound for the nullities of the ( k + 1 ) -line graphs of a particular family of k-trees, generalizing a known result about the nullities of line graphs of trees. We also present a general characterization for the ( k + 1 ) -line graphs of k-trees.

In this paper, we study global bifurcation diagrams and the exact multiplicity of positive solutions for Minkowski-curvature problem $$\begin{aligned} \left\{ \begin{array}{l} -\left( u^{\prime }/\sqrt{1-{u^{\prime }}^{2}}\right) ^{\prime }=\lambda \exp u,\text { in }\left( -L,L\right) , \\ u(-L)=u(L)=0, \end{array} \right. \end{aligned}$$ where $$\lambda >0$$ is a bifurcation parameter and $$L>0$$ is an evolution parameter. It can be viewed as a variant of the one-dimensional Liouville–Bratu–Gelfand problem. We prove that there exists $$L_{0}>0$$ such that the bifurcation curve $$S_{L}$$ is S-shaped for $$L>L_{0}$$ and is monotone increasing for $$0

In this paper we study the existence and the multiplicity of nontrivial weak solutions for a fourth order variable exponent Kirchhoff type problem involving p(x)-biharmonic operator with changing sign weight and with no flux boundary condition. By using variational approach and the theory of variable exponent Sobolev spaces, we determine an interval of parameters for which this problem admits at least two nontrivial weak solutions.

In this paper, a numerical scheme based on the Galerkin method is extended for solving one-dimensional hyperbolic partial differential equations with a nonlocal conservation condition. To achieve this goal, we apply the interpolating scaling functions. The most important advantages of these bases are orthonormality, interpolation, and having flexible vanishing moments. In other words, to increase the accuracy of the approximation, we can individually or simultaneously increase both the degree of polynomials (multiplicity r ) and the level of refinement (refinement level J ). The convergence analysis is investigated, and numerical examples guarantee it. To show the ability of the proposed method, we compare it with existing methods, and it can be confirmed that our results are better than them.

We deal with the existence and multiplicity of solutions for a non-local fractional system involving combined critical non-linearities. Some existence results are obtained by splitting the Nehari manifold and applying the Mountain–Pass theorem of Ambrosetti and Rabinowitz.

Let G be a simple connected graph and A ( G ) be the adjacency matrix of G. The graph G is said to have the reciprocal eigenvalue property (R) if A ( G ) is nonsingular and 1 λ is an eigenvalue of A ( G ) whenever λ is an eigenvalue of A ( G ) . Further, if λ and 1 λ have the same multiplicity, for each eigenvalue λ, then G is said to have the strong reciprocal eigenvalue property (SR). Till date, all the classes of bipartite graphs that are found to have property (R), are found to have property (SR) and it is not known whether these two properties are equivalent even for the bipartite graphs with a unique perfect matching. Among nonbipartite graphs, there is only one known graph class for which these two properties are not equivalent. In this article, we construct some more classes of nonbipartite graphs with property (R) but not (SR).

In this paper, we focus on the standing waves with prescribed mass for the Schrodinger equations with van der Waals type potentials, that is, two-body potentials with different width. This leads to the study of the following nonlocal elliptic equation − Δ u = λ u + μ ( | x | − α ⁎ | u | 2 ) u + ( | x | − β ⁎ | u | 2 ) u , x ∈ R N under the normalized constraint ∫ R N u 2 = c > 0 , where N ≥ 3 , μ > 0 , α, β ∈ ( 0 , N ) , and the frequency λ ∈ R is unknown and appears as Lagrange multiplier. Compared with the well studied case α = β , the solution set of the above problem with different width of two body potentials α ≠ β is much richer. Under different assumptions on c, α and β, we prove several existence, multiplicity and asymptotic behavior of solutions to the above problem. In addition, the stability of the corresponding standing waves for the related time-dependent problem is discussed.

We prove Clifford theoretic results on the representations of finite groups which only hold in characteristic $2$. Let $G$ be a finite group, let $N$ be a normal subgroup of $G$ and let $\varphi$ be an irreducible $2$-Brauer character of $N$ which is self-dual. We prove that there is a unique self-dual irreducible Brauer character $\theta$ of $G$ such that $\varphi$ occurs with odd multiplicity in the restriction of $\theta$ to $N$. Moreover this multiplicity is $1$. Conversely if $\theta$ is an irreducible $2$-Brauer character of $G$ which is self-dual but not of quadratic type, the restriction of $\theta$ to $N$ is a sum of distinct self-dual irreducible Brauer character of $N$, none of which have quadratic type. Let $b$ be a real $2$-block of $N$. We show that there is a unique real $2$-block of $G$ covering $b$ which is weakly regular., Comment: 19 pages, example added

In this work we investigate the following fractional p-Laplacian differential equation with Sturm–Liouville boundary value conditions: $$ \textstyle\begin{cases} {}_{t}D_{T}^{\alpha } ( { \frac{1}{{{{ ( {h ( t )} )}^{p - 2}}}}{\phi _{p}} ( {h ( t ){}_{0}^{C}D_{t}^{\alpha }u ( t )} )} ) + a ( t ){\phi _{p}} ( {u ( t )} ) = \lambda f (t,u(t) ),\quad \mbox{a.e. }t \in [ {0,T} ], \\ {\alpha _{1}} {\phi _{p}} ( {u ( 0 )} ) - { \alpha _{2}} {}_{t}D_{T}^{\alpha - 1} ( {{\phi _{p}} ( {{}_{0}^{C}D_{t}^{\alpha }u ( 0 )} )} ) = 0, \\ {\beta _{1}} { \phi _{p}} ( {u ( T )} ) + {\beta _{2}} {}_{t}D_{T}^{ \alpha - 1} ( {{\phi _{p}} ( {{}_{0}^{C}D_{t}^{\alpha }u ( T )} )} ) = 0, \end{cases} $$ { D T α t ( 1 ( h ( t ) ) p − 2 ϕ p ( h ( t ) 0 C D t α u ( t ) ) ) + a ( t ) ϕ p ( u ( t ) ) = λ f ( t , u ( t ) ) , a.e. t ∈ [ 0 , T ] , α 1 ϕ p ( u ( 0 ) ) − α 2 t D T α − 1 ( ϕ p ( 0 C D t α u ( 0 ) ) ) = 0 , β 1 ϕ p ( u ( T ) ) + β 2 t D T α − 1 ( ϕ p ( 0 C D t α u ( T ) ) ) = 0 , where ${}_{0}^{C}D_{t}^{\alpha }$ D t α 0 C , ${}_{t}D_{T}^{\alpha }$ D T α t are the left Caputo and right Riemann–Liouville fractional derivatives of order $\alpha \in ( {\frac{1}{2},1} ]$ α ∈ ( 1 2 , 1 ] , respectively. By using variational methods and critical point theory, some new results on the multiplicity of solutions are obtained.

We obtain the multiplicity of solutions for the following Schr $$\ddot{\text{ o }}$$ dinger equations $$\begin{aligned} \left\{ \begin{array}{l@{\quad }l} -\Delta u+V(x)u=g(x,u)&{}\text{ for } x\in {\mathbb {R}}^{N},\\ u(x)\rightarrow 0&{}\text{ as } |u|\rightarrow \infty , \end{array} \right. \end{aligned}$$ where $$V\in C({\mathbb {R}}^{N},{\mathbb {R}})$$ is coercive at infinity and g involves concave–convex nonlinearities while the convex terms need not to satisfy the (AR)-type condition. Some new nonlinearities are considered and an example is given.

In this paper, we are concerned with the following nonlinear magnetic Schrodinger equation with critical growth: $$\left\{ {\matrix{ {{{\left( {{\varepsilon \over i}\nabla - A\left( x \right)} \right)}^2}u + V\left( x \right)u = f\left( {{{\left| u \right|}^2}} \right)u + {{\left| u \right|}^{{2^*} - 2}}u\,\,\,\,\,\,{\rm{in}}\,{\mathbb{R}^N},} \hfill \cr {u \in {H^1}\left( {{\mathbb{R}^N},\mathbb{C}} \right),} \hfill \cr } } \right.$$ where ∊ > 0 is a parameter, N ≥ 3 and $${2^*} = {{2N} \over {N - 2}}$$ is the Sobolev critical exponent, V: ℝN → ℝ and A: ℝN → ℝN are continuous potentials, f: ℝ → ℝ is a subcritical nonlinear term. Under a local assumption on the potential V, by the variational methods, the penalization technique and the Ljusternic—Schnirelmann theory, we prove the multiplicity and concentration of nontrivial solutions of the above problem for e small. For the problem, the function f is only continuous, which allows to consider larger classes of nonlinearities in the reaction.

In this paper, we investigate the existence of an S-shaped connected component in the set of positive solutions for a Minkowski-curvature Dirichlet problem with indefinite weight. By figuring the shape of unbounded continua of solutions, we show the existence and multiplicity of positive solutions with respect to the parameter $$\lambda $$ . In particular, we obtain the existence of at least three positive solutions for $$\lambda $$ being in a certain interval.

For two random vectors whose dimensions are both proportional to the sample size, we in this paper propose two ridge ratio criteria to determine the number of canonical correlation pairs. The criteria are, respectively, based on eigenvalue difference-based and centered eigenvalue-based ridge ratios. Unlike existing methods, the criteria make the ratio at the index we want to identify stick out to show a visualized “valley-cliff” pattern and thus can adequately avoid the local optimal solutions that often occur in the eigenvalues multiplicity cases. The numerical studies also suggest its advantage over existing scree plot-based method that is not a visualization method and more seriously underestimates the number of pairs than the proposed ones and the AIC and $$C_p$$ criteria that often extremely over-estimate the number, and the BIC criterion that has very serious underestimation problem. A real data set is analyzed for illustration.

Let $$\mathcal {P}_r$$ denote an almost-prime with at most r prime factors, counted according to multiplicity. In this paper, we generalize the result of Vaughan (Bull Lond Math Soc 17(1):17–20, 1985) for ternary ‘admissible exponent’. Moreover, we use the refined ‘admissible exponent’ to prove that, for $$3\leqslant k\leqslant 14$$ and for every sufficiently large even integer n, the following equation $$\begin{aligned} n=x^2+p_1^2+p_2^3+p_3^3+p_4^3+p_5^k \end{aligned}$$ is solvable with x being an almost-prime $$\mathcal {P}_{r(k)}$$ and the other variables primes, where r(k) is defined in Theorem 1.1. This result constitutes a deepening of previous results.

The puzzling observation that the famous Euler's formula for three-dimensional polyhedra V − E + F = 2 or Euler characteristic χ = V − E + F − I = 1 (where V, E, F are the numbers of the bounding vertices, edges and faces, respectively, and I = 1 counts the single solid itself) when applied to space-filling solids, such as crystallographic asymmetric units or Dirichlet domains, are modified in such a way that they sum up to a value one unit smaller (i.e. to 1 or 0, respectively) is herewith given general validity. The proof provided in this paper for the modified Euler characteristic, χm = V m − E m + F m − I m = 0, is divided into two parts. First, it is demonstrated for translational lattices by using a simple argument based on parity groups of integer-indexed elements of the lattice. Next, Whitehead's theorem, about the invariance of the Euler characteristic, is used to extend the argument from the unit cell to its asymmetric unit components.

Erdős proved that F ( A ) : = ∑ a ∈ A 1 a log ⁡ a converges for any primitive set of integers A and later conjectured this sum is maximized when A is the set of primes. Banks and Martin further conjectured that F ( P 1 ) > ⋯ > F ( P k ) > F ( P k + 1 ) > ⋯ , where P j is the set of integers with j prime factors counting multiplicity, though this was recently disproven by Lichtman. We consider the corresponding problems over the function field F q [ x ] , investigating the sum F ( A ) : = ∑ f ∈ A 1 deg f ⋅ q deg f . We establish a uniform bound for F ( A ) over all primitive sets of polynomials A ⊂ F q [ x ] and conjecture that it is maximized by the set of monic irreducible polynomials. We find that the analogue of the Banks-Martin conjecture is false for q = 2 , 3, and 4, but we find computational evidence that it holds for q > 4 .

We present a general formulation of the averaging method in the setting of a semilinear equation L x = e N ( x , e ) , being L a linear Fredholm mapping of index zero. Our general approach provides new results even in the classical periodic framework. Among the applications we obtained there are: a partial answer to an open problem related to the Liebau phenomenon, the multiplicity of periodic solutions for a planar system with delay and the existence of solution for a nonlocal chemical reactor.

In this paper, we study some results on the existence and multiplicity of solutions for a class of nonlocal quasilinear elliptic systems. In fact, we prove the existence of precise intervals of positive parameters such that the problem admits multiple solutions. Our approach is based on variational methods.

In this paper, we study a class of Hamiltonian elliptic system. Under some assumptions for the nonlinearities, we obtain the existence of sign-changing solutions of this system and the multiplicity...

We are concerned with the existence and multiplicity of nontrivial solutions to the following double phase problems: $$\begin{aligned} \left\{ \begin{array}{ll} -\mathrm{div}(|\nabla u|^{p-2}\nabla u+\alpha (x)|\nabla u|^{q-2}\nabla u)+V(x)|u|^{\gamma -2}u=f(x,u),&{}\ \mathrm{in}\ \Omega ,\\ u=0,&{}\ \mathrm{on}\ {\partial \Omega ,} \end{array}\right. \end{aligned}$$ applying the mountain pass theorem and fountain theorem. The Ambrosetti—Rabinowitz condition as well as the monotonicity of $$f(x,t)/|t|^{q-1}$$ are not assumed.

In this paper, we investigate the multiplicity of solutions for Kirchhoff fractional p-Laplacian system in bounded domains: ∑ i = 1 k [ u i ] s , p p θ − 1 ( − Δ ) p s u j ( x ) = λ j f j ( x ) | u...

The present paper is concerned with the half-space Dirichlet problem "Equation missing"where $$\mathbb {R}^{N}_{+}:= \{\,x \in \mathbb {R}^N: x_N > 0\, \}$$ R + N : = { x ∈ R N : x N > 0 } for some $$N \ge 1$$ N ≥ 1 and $$p > 1$$ p > 1 , $$c > 0$$ c > 0 are constants. We analyse the existence, non-existence and multiplicity of bounded positive solutions to ($$P_c$$ P c ). We prove that the existence and multiplicity of bounded positive solutions to ($$P_c$$ P c ) depend in a striking way on the value of $$c > 0$$ c > 0 and also on the dimension N. We find an explicit number $${c_p}\in (1,\sqrt{e})$$ c p ∈ ( 1 , e ) , depending only on p, which determines the threshold between existence and non-existence. In particular, in dimensions $$N \ge 2$$ N ≥ 2 , we prove that, for $$0< c < {c_p}$$ 0 < c < c p , problem ($$P_c$$ P c ) admits infinitely many bounded positive solutions, whereas, for $$c > {c_p}$$ c > c p , there are no bounded positive solutions to ($$P_c$$ P c ).

We provide sufficient conditions for the existence of one positive solution for a fourth--order beam equation with a discontinuous nonlinear term. Also a multiplicity result is established. They are based on a recent generalization of the Krasnosel’skiĭ fixed point theorem in cones.

In the paper, we study a class of semilinear fractional semilinear elliptic equations involving concave-convex nonlinearities: \begin{document}$ \begin{equation*} \left\{ \begin{array}{ll} (-\Delta)^{\alpha} u+V_{\lambda }\left( x\right) u = f\left( x\right) \left\vert u\right\vert ^{q-2}u+g\left( x\right) \left\vert u\right\vert ^{p-2}u & \text{in }\mathbb{R}^{N}, \\ u\in H^{\alpha}(\mathbb{R}^{N}), & \end{array}\right. \end{equation*} $\end{document} where \begin{document}$ \alpha\in (0,1] $\end{document} , \begin{document}$ 1 2\alpha\right), $\end{document} the potential \begin{document}$ V_{\lambda }(x) = \lambda a(x)-b(x) $\end{document} and the parameter \begin{document}$ \lambda >0. $\end{document} Under some suitable assumptions on \begin{document}$ a,b $\end{document} and the weight functions \begin{document}$ f,g $\end{document} , we obtain the existence and multiplicity of non-trivial (positive) solutions for \begin{document}$ \lambda $\end{document} large enough. An interesting phenomenon is that we do not need the condition that weight functions \begin{document}$ f, g $\end{document} are integrable or bounded on whole space \begin{document}$ \mathbb{R}^{N}. $\end{document}

By using the Krasnoselskii fixed point theorem, sufficient conditions are obtained for the existence and multiplicity of positive periodic solutions for a class of second order damped functional differential equations with multiple delays. Our results are a further expansion of the previous research results.

In this paper, we deal with a coupled system of nonlinear fractional multi-point boundary value problems with $ p $-Laplacian operator. The existence and multiplicity of positive solutions are obtained by employing Leray-Schauder alternative theory, Leggett-Williams fixed point theorem and Avery-Henderson fixed point theorem. As an application, two examples are given to illustrate the effectiveness of our main results.

This paper establishes global bifurcation and eigenvalue results for the following complex $ k $-Hessian equation $ \begin{equation} \left\{ \begin{array}{ll} S_k\left(u_{i\overline{j}}\right) = \lambda^k f(-u)\,\, &\text{in}\,\, B,\\ u = 0 &\text{on}\,\, \partial B. \end{array} \right.\nonumber \end{equation} $ The existence/nonexistence, uniqueness and multiplicity of radially symmetric negative solutions are investigated. Moreover, a-priori bound of radially symmetric negative solutions is also obtained.