text:
Canadian Mathematical Society www.cms.math.ca
 location:  Publications → journals
Search results

Search: All articles in the CMB digital archive with keyword space

 Expand all        Collapse all Results 1 - 25 of 131

1. CMB Online first

Christ, Michael; Rieffel, Marc A.
 Nilpotent group C*-algebras as compact quantum metric spaces Let $\mathbb{L}$ be a length function on a group $G$, and let $M_\mathbb{L}$ denote the operator of pointwise multiplication by $\mathbb{L}$ on $\lt(G)$. Following Connes, $M_\mathbb{L}$ can be used as a Dirac'' operator for the reduced group C*-algebra $C_r^*(G)$. It defines a Lipschitz seminorm on $C_r^*(G)$, which defines a metric on the state space of $C_r^*(G)$. We show that for any length function satisfying a strong form of polynomial growth on a discrete group, the topology from this metric coincides with the weak-$*$ topology (a key property for the definition of a compact quantum metric space''). In particular, this holds for all word-length functions on finitely generated nilpotent-by-finite groups. Keywords:group C*-algebra, Dirac operator, quantum metric space, discrete nilpotent group, polynomial growthCategories:46L87, 20F65, 22D15, 53C23, 58B34

2. CMB Online first

Ghaani Farashahi, Arash
 Abstract Plancherel (Trace) Formulas over Homogeneous Spaces of Compact Groups This paper introduces a unified operator theory approach to the abstract Plancherel (trace) formulas over homogeneous spaces of compact groups. Let $G$ be a compact group and $H$ be a closed subgroup of $G$. Let $G/H$ be the left coset space of $H$ in $G$ and $\mu$ be the normalized $G$-invariant measure on $G/H$ associated to the Weil's formula. Then, we present a generalized abstract notion of Plancherel (trace) formula for the Hilbert space $L^2(G/H,\mu)$. Keywords:compact group, homogeneous space, dual space, Plancherel (trace) formulaCategories:20G05, 43A85, 43A32, 43A40

3. CMB Online first

Mihăilescu, Mihai; Moroşanu, Gheorghe
 Eigenvalues of $-\Delta_p -\Delta_q$ under Neumann boundary condition The eigenvalue problem $-\Delta_p u-\Delta_q u=\lambda|u|^{q-2}u$ with $p\in(1,\infty)$, $q\in(2,\infty)$, $p\neq q$ subject to the corresponding homogeneous Neumann boundary condition is investigated on a bounded open set with smooth boundary from $\mathbb{R}^N$ with $N\geq 2$. A careful analysis of this problem leads us to a complete description of the set of eigenvalues as being a precise interval $(\lambda_1, +\infty )$ plus an isolated point $\lambda =0$. This comprehensive result is strongly related to our framework which is complementary to the well-known case $p=q\neq 2$ for which a full description of the set of eigenvalues is still unavailable. Keywords:eigenvalue problem, Sobolev space, Nehari manifold, variational methodsCategories:35J60, 35J92, 46E30, 49R05

4. CMB Online first

Bačák, Miroslav; Kovalev, Leonid V.
 Lipschitz retractions in Hadamard spaces via gradient flow semigroups Let $X(n),$ for $n\in\mathbb{N},$ be the set of all subsets of a metric space $(X,d)$ of cardinality at most $n.$ The set $X(n)$ equipped with the Hausdorff metric is called a finite subset space. In this paper we are concerned with the existence of Lipschitz retractions $r\colon X(n)\to X(n-1)$ for $n\ge2.$ It is known that such retractions do not exist if $X$ is the one-dimensional sphere. On the other hand L. Kovalev has recently established their existence in case $X$ is a Hilbert space and he also posed a question as to whether or not such Lipschitz retractions exist for $X$ being a Hadamard space. In the present paper we answer this question in the positive. Keywords:finite subset space, gradient flow, Hadamard space, Lie-Trotter-Kato formula, Lipschitz retractionCategories:53C23, 47H20, 54E40, 58D07

5. CMB Online first

Liao, Fanghui; Liu, Zongguang
 Some Properties of Triebel-Lizorkin and Besov Spaces Associated with Zygmund Dilations In this paper, using CalderÃ³n's reproducing formula and almost orthogonality estimates, we prove the lifting property and the embedding theorem of the Triebel-Lizorkin and Besov spaces associated with Zygmund dilations. Keywords:Triebel-Lizorkin and Besov spaces, Riesz potential, CalderÃ³n's reproducing formula, almost orthogonality estimate, Zygmund dilation, embedding theoremCategories:42B20, 42B35

6. CMB Online first

García-Pacheco, Francisco Javier; Hill, Justin R.
 Geometric characterizations of Hilbert spaces We study some geometric properties related to the set $\Pi_X:= \{ (x,x^* )\in\mathsf{S}_X\times \mathsf{S}_{X^*}:x^* (x )=1 \}$ obtaining two characterizations of Hilbert spaces in the category of Banach spaces. We also compute the distance of a generic element $(h,k )\in H\oplus_2 H$ to $\Pi_H$ for $H$ a Hilbert space. Keywords:Hilbert space, extreme point, smooth, $\mathsf{L}^2$-summandsCategories:46B20, 46C05

7. CMB Online first

wang, jianfei
 The Carleson measure problem between analytic Morrey spaces The purpose of this paper is to characterize positive measure $\mu$ on the unit disk such that the analytic Morrey space $\mathcal{AL}_{p,\eta}$ is boundedly and compactly embedded to the tent space $\mathcal{T}_{q,1-\frac{q}{p}(1-\eta)}^{\infty}(\mu)$ for the case $1\leq q\leq p\lt \infty$ respectively. As an application, these results are used to establish the boundedness and compactness of integral operators and multipliers between analytic Morrey spaces. Keywords:Morrey space, Carleson measure problem, boundedness, compactnessCategories:30H35, 28A12, 47B38, 46E15

8. CMB 2015 (vol 58 pp. 692)

Anona, F. M.; Randriambololondrantomalala, Princy; Ravelonirina, H. S. G.
 Sur les algÃ¨bres de Lie associÃ©es Ã  une connexion Let $\Gamma$ be a connection on a smooth manifold $M$, in this paper we give some properties of $\Gamma$ by studying the corresponding Lie algebras. In particular, we compute the first Chevalley-Eilenberg cohomology space of the horizontal vector fields Lie algebra on the tangent bundle of $M$, whose the corresponding Lie derivative of $\Gamma$ is null, and of the horizontal nullity curvature space. Keywords:algÃ¨bre de Lie, connexion, cohomologie de Chevalley-Eilenberg, champs dont la dÃ©rivÃ©e de Lie correspondante Ã  une connexion est nulle, espace de nullitÃ© de la courbureCategories:17B66, 53B15

9. CMB 2015 (vol 59 pp. 104)

He, Ziyi; Yang, Dachun; Yuan, Wen
 Littlewood-Paley Characterizations of Second-Order Sobolev Spaces via Averages on Balls In this paper, the authors characterize second-order Sobolev spaces $W^{2,p}({\mathbb R}^n)$, with $p\in [2,\infty)$ and $n\in\mathbb N$ or $p\in (1,2)$ and $n\in\{1,2,3\}$, via the Lusin area function and the Littlewood-Paley $g_\lambda^\ast$-function in terms of ball means. Keywords:Sobolev space, ball means, Lusin-area function, $g_\lambda^*$-functionCategories:46E35, 42B25, 42B20, 42B35

10. CMB 2015 (vol 59 pp. 3)

Alfuraidan, Monther Rashed
 The Contraction Principle for Multivalued Mappings on a Modular Metric Space with a Graph We study the existence of fixed points for contraction multivalued mappings in modular metric spaces endowed with a graph. The notion of a modular metric on an arbitrary set and the corresponding modular spaces, generalizing classical modulars over linear spaces like Orlicz spaces, were recently introduced. This paper can be seen as a generalization of Nadler's and Edelstein's fixed point theorems to modular metric spaces endowed with a graph. Keywords:fixed point theory, modular metric spaces, multivalued contraction mapping, connected digraph.Categories:47H09, 46B20, 47H10, 47E10

11. CMB 2015 (vol 58 pp. 757)

Han, Yanchang
 Embedding Theorem for Inhomogeneous Besov and Triebel-Lizorkin Spaces on RD-spaces In this article we prove the embedding theorem for inhomogeneous Besov and Triebel-Lizorkin spaces on RD-spaces. The crucial idea is to use the geometric density condition on the measure. Keywords:spaces of homogeneous type, test function space, distributions, CalderÃ³n reproducing formula, Besov and Triebel-Lizorkin spaces, embeddingCategories:42B25, 46F05, 46E35

12. CMB 2015 (vol 58 pp. 459)

Casini, Emanuele; Miglierina, Enrico; Piasecki, Lukasz
 Hyperplanes in the Space of Convergent Sequences and Preduals of $\ell_1$ The main aim of the present paper is to investigate various structural properties of hyperplanes of $c$, the Banach space of the convergent sequences. In particular, we give an explicit formula for the projection constants and we prove that an hyperplane of $c$ is isometric to the whole space if and only if it is $1$-complemented. Moreover, we obtain the classification of those hyperplanes for which their duals are isometric to $\ell_{1}$ and we give a complete description of the preduals of $\ell_{1}$ under the assumption that the standard basis of $\ell_{1}$ is weak$^{*}$-convergent. Keywords:space of convergent sequences, projection, $\ell_1$-predual, hyperplaneCategories:46B45, 46B04

13. CMB 2015 (vol 58 pp. 596)

Ongaro, Jared; Shapiro, Boris
 A Note on Planarity Stratification of Hurwitz Spaces One can easily show that any meromorphic function on a complex closed Riemann surface can be represented as a composition of a birational map of this surface to $\mathbb{CP}^2$ and a projection of the image curve from an appropriate point $p\in \mathbb{CP}^2$ to the pencil of lines through $p$. We introduce a natural stratification of Hurwitz spaces according to the minimal degree of a plane curve such that a given meromorphic function can be represented in the above way and calculate the dimensions of these strata. We observe that they are closely related to a family of Severi varieties studied earlier by J. Harris, Z. Ran and I. Tyomkin. Keywords:Hurwitz spaces, meromorphic functions, Severi varieties

14. CMB 2015 (vol 58 pp. 271)

Jafari, Sayyed Heidar; Jafari Rad, Nader
 On Domination of Zero-divisor Graphs of Matrix Rings We study domination in zero-divisor graphs of matrix rings over a commutative ring with $1$. Keywords:vector space, linear transformation, zero-divisor graph, domination, local ringCategory:05C69

15. CMB 2015 (vol 58 pp. 350)

Merino-Cruz, Héctor; Wawrzynczyk, Antoni
 On Closed Ideals in a Certain Class of Algebras of Holomorphic Functions We recently introduced a weighted Banach algebra $\mathfrak{A}_G^n$ of functions which are holomorphic on the unit disc $\mathbb{D}$, continuous up to the boundary and of the class $C^{(n)}$ at all points where the function $G$ does not vanish. Here, $G$ refers to a function of the disc algebra without zeros on $\mathbb{D}$. Then we proved that all closed ideals in $\mathfrak{A}_G^n$ with at most countable hull are standard. In the present paper, on the assumption that $G$ is an outer function in $C^{(n)}(\overline{\mathbb{D}})$ having infinite roots in $\mathfrak{A}_G^n$ and countable zero set $h(G)$, we show that all the closed ideals $I$ with hull containing $h(G)$ are standard. Keywords:Banach algebra, disc algebra, holomorphic spaces, standard idealCategories:46J15, 46J20, 30H50

16. CMB 2015 (vol 58 pp. 507)

Hsu, Ming-Hsiu; Lee, Ming-Yi
 VMO Space Associated with Parabolic Sections and its Application In this paper we define $VMO_\mathcal{P}$ space associated with a family $\mathcal{P}$ of parabolic sections and show that the dual of $VMO_\mathcal{P}$ is the Hardy space $H^1_\mathcal{P}$. As an application, we prove that almost everywhere convergence of a bounded sequence in $H^1_\mathcal{P}$ implies weak* convergence. Keywords:Monge-Ampere equation, parabolic section, Hardy space, BMO, VMOCategory:42B30

17. CMB 2015 (vol 58 pp. 808)

Liu, Feng; Wu, Huoxiong
 On the Regularity of the Multisublinear Maximal Functions This paper is concerned with the study of the regularity for the multisublinear maximal operator. It is proved that the multisublinear maximal operator is bounded on first-order Sobolev spaces. Moreover, two key point-wise inequalities for the partial derivatives of the multisublinear maximal functions are established. As an application, the quasi-continuity on the multisublinear maximal function is also obtained. Keywords:regularity, multisublinear maximal operator, Sobolev spaces, partial deviative, quasicontinuityCategories:42B25, 46E35

18. CMB 2015 (vol 58 pp. 241)

Botelho, Fernanda
 Isometries and Hermitian Operators on Zygmund Spaces In this paper we characterize the isometries of subspaces of the little Zygmund space. We show that the isometries of these spaces are surjective and represented as integral operators. We also show that all hermitian operators on these settings are bounded. Keywords:Zygmund spaces, the little Zygmund space, Hermitian operators, surjective linear isometries, generators of one-parameter groups of surjective isometriesCategories:46E15, 47B15, 47B38

19. CMB 2014 (vol 58 pp. 432)

Yang, Dachun; Yang, Sibei
 Second-order Riesz Transforms and Maximal Inequalities Associated with Magnetic SchrÃ¶dinger Operators Let $A:=-(\nabla-i\vec{a})\cdot(\nabla-i\vec{a})+V$ be a magnetic SchrÃ¶dinger operator on $\mathbb{R}^n$, where $\vec{a}:=(a_1,\dots, a_n)\in L^2_{\mathrm{loc}}(\mathbb{R}^n,\mathbb{R}^n)$ and $0\le V\in L^1_{\mathrm{loc}}(\mathbb{R}^n)$ satisfy some reverse HÃ¶lder conditions. Let $\varphi\colon \mathbb{R}^n\times[0,\infty)\to[0,\infty)$ be such that $\varphi(x,\cdot)$ for any given $x\in\mathbb{R}^n$ is an Orlicz function, $\varphi(\cdot,t)\in {\mathbb A}_{\infty}(\mathbb{R}^n)$ for all $t\in (0,\infty)$ (the class of uniformly Muckenhoupt weights) and its uniformly critical upper type index $I(\varphi)\in(0,1]$. In this article, the authors prove that second-order Riesz transforms $VA^{-1}$ and $(\nabla-i\vec{a})^2A^{-1}$ are bounded from the Musielak-Orlicz-Hardy space $H_{\varphi,\,A}(\mathbb{R}^n)$, associated with $A$, to the Musielak-Orlicz space $L^{\varphi}(\mathbb{R}^n)$. Moreover, the authors establish the boundedness of $VA^{-1}$ on $H_{\varphi, A}(\mathbb{R}^n)$. As applications, some maximal inequalities associated with $A$ in the scale of $H_{\varphi, A}(\mathbb{R}^n)$ are obtained. Keywords:Musielak-Orlicz-Hardy space, magnetic SchrÃ¶dinger operator, atom, second-order Riesz transform, maximal inequalityCategories:42B30, 42B35, 42B25, 35J10, 42B37, 46E30

20. CMB 2014 (vol 58 pp. 150)

Ostrovskii, Mikhail I.
 Connections Between Metric Characterizations of Superreflexivity and the Radon-NikodÃ½ Property for Dual Banach Spaces Johnson and Schechtman (2009) characterized superreflexivity in terms of finite diamond graphs. The present author characterized the Radon-NikodÃ½m property (RNP) for dual spaces in terms of the infinite diamond. This paper is devoted to further study of relations between metric characterizations of superreflexivity and the RNP for dual spaces. The main result is that finite subsets of any set $M$ whose embeddability characterizes the RNP for dual spaces, characterize superreflexivity. It is also observed that the converse statement does not hold, and that $M=\ell_2$ is a counterexample. Keywords:Banach space, diamond graph, finite representability, metric characterization, Radon-NikodÃ½m property, superreflexivityCategories:46B85, 46B07, 46B22

21. CMB 2014 (vol 58 pp. 158)

Özgür, Cihan; Mihai, Adela
 Corrigendum to "Chen Inequalities for Submanifolds of Real Space Forms with a Semi-symmetric Non-metric Connection" We fix the coefficients in the inequality (4.1) in the Theorem 4.1(i) from A. Mihai and C. ÃzgÃ¼r, "Chen inequalities for submanifolds of real space forms with a semi-symmetric non-metric connection" Canad. Math. Bull. 55 (2012), no. 3, 611-622. Keywords:real space form, semi-symmetric non-metric connection, Ricci curvatureCategories:53C40, 53B05, 53B15

22. CMB 2014 (vol 58 pp. 128)

Marković, Marijan
 A Sharp Constant for the Bergman Projection For the Bergman projection operator $P$ we prove that \begin{equation*} \|P\colon L^1(B,d\lambda)\rightarrow B_1\| = \frac {(2n+1)!}{n!}. \end{equation*} Here $\lambda$ stands for the hyperbolic metric in the unit ball $B$ of $\mathbb{C}^n$, and $B_1$ denotes the Besov space with an adequate semi--norm. We also consider a generalization of this result. This generalizes some recent results due to PerÃ¤lÃ¤. Keywords:Bergman projections, Besov spacesCategories:45P05, 47B35

23. CMB 2014 (vol 57 pp. 749)

Cavalieri, Renzo; Marcus, Steffen
 Geometric Perspective on Piecewise Polynomiality of Double Hurwitz Numbers We describe double Hurwitz numbers as intersection numbers on the moduli space of curves $\overline{\mathcal{M}}_{g,n}$. Using a result on the polynomiality of intersection numbers of psi classes with the Double Ramification Cycle, our formula explains the polynomiality in chambers of double Hurwitz numbers, and the wall crossing phenomenon in terms of a variation of correction terms to the $\psi$ classes. We interpret this as suggestive evidence for polynomiality of the Double Ramification Cycle (which is only known in genera $0$ and $1$). Keywords:double Hurwitz numbers, wall crossings, moduli spaces, ELSV formulaCategory:14N35

24. CMB 2014 (vol 58 pp. 297)

Khamsi, M. A.
 Approximate Fixed Point Sequences of Nonlinear Semigroup in Metric Spaces In this paper, we investigate the common approximate fixed point sequences of nonexpansive semigroups of nonlinear mappings $\{T_t\}_{t \geq 0}$, i.e., a family such that $T_0(x)=x$, $T_{s+t}=T_s(T_t(x))$, where the domain is a metric space $(M,d)$. In particular we prove that under suitable conditions, the common approximate fixed point sequences set is the same as the common approximate fixed point sequences set of two mappings from the family. Then we use the Ishikawa iteration to construct a common approximate fixed point sequence of nonexpansive semigroups of nonlinear mappings. Keywords:approximate fixed point, fixed point, hyperbolic metric space, Ishikawa iterations, nonexpansive mapping, semigroup of mappings, uniformly convex hyperbolic spaceCategories:47H09, 46B20, 47H10, 47E10

25. CMB 2014 (vol 57 pp. 803)

Gabriyelyan, S. S.
 Free Locally Convex Spaces and the $k$-space Property Let $L(X)$ be the free locally convex space over a Tychonoff space $X$. Then $L(X)$ is a $k$-space if and only if $X$ is a countable discrete space. We prove also that $L(D)$ has uncountable tightness for every uncountable discrete space $D$. Keywords:free locally convex space, $k$-space, countable tightnessCategories:46A03, 54D50, 54A25
 Page 1 2 3 4 ... 6 Next
 top of page | contact us | privacy | site map |

© Canadian Mathematical Society, 2016 : https://cms.math.ca/


text:
Research

# Existence of solutions for a general quasilinear elliptic system via perturbation method

Yujuan Jiao1*, Shengmao Fu2 and Yanli Wang3

Author Affiliations

1 College of Mathematics and Computer Science, Northwest University for Nationalities, Lanzhou, 730124, P.R. China

2 College of Mathematics and Statistics, Northwest Normal University, Lanzhou, 730070, P.R. China

3 School of Mathematical Sciences, Beijing Normal University, Beijing, 100875, P.R. China

For all author emails, please log on.

Boundary Value Problems 2013, 2013:219  doi:10.1186/1687-2770-2013-219

The electronic version of this article is the complete one and can be found online at: http://www.boundaryvalueproblems.com/content/2013/1/219

 Received: 28 May 2013 Accepted: 28 August 2013 Published: 7 November 2013

© 2013 Jiao et al.; licensee Springer.

This is an open access article distributed under the terms of the Creative Commons Attribution License (http://creativecommons.org/licenses/by/2.0), which permits unrestricted use, distribution, and reproduction in any medium, provided the original work is properly cited.

### Abstract

In this paper, we consider the following quasilinear elliptic system:

where , , , , , , is the critical Sobolev exponent and () is a bounded smooth domain. By using the perturbation method, we establish the existence of both positive and negative solutions for this system.

MSC: 35J60, 35B33.

##### Keywords:
quasilinear elliptic system; positive solution; negative solution; perturbation method

### 1 Introduction

Let us consider the following quasilinear elliptic system:

(1.1)

where , , , , , , is the critical Sobolev exponent and () is a bounded smooth domain. This system includes the following special class of system with , , i.e.,

which is referred to as the so-called modified nonlinear Schrödinger system.

Our assumptions on the functions and are as follows.

(A1) The functions , , , , .

(A2) There exist constants , , , satisfying , , and such that

for , , .

(A3)

for , , .

In recent years, much attention has been devoted to the quasilinear Schrödinger equation of the following form:

(1.2)

See, for example, [1] where Poppenberg et al. proved the existence of a positive ground state solution by using a constrained minimization argument. Using a change of variables, Liu et al.[2] used an Orlicz space to prove the existence of a soliton solution for equation (1.2) via the mountain pass theorem. Colin and Jeanjean [3] also made use of a change of variables but worked in the Sobolev space . They proved the existence of a positive solution for equation (1.2) from the classical results given by Berestycki and Lions [4]. Liu et al.[5] established the existence of both one-sign and nodal ground states of soliton-type solutions for equation (1.2) by the Nehari method. By using the Nehari manifold method and the concentration compactness principle (see [6]) in the Orlicz space, Guo and Tang [7] considered the following quasilinear Schrödinger system:

(1.3)

with , having a potential well and , , , and they proved the existence of a ground state solution for system (1.3) which localizes near the potential well for λ large enough. Guo and Tang [8] considered also ground state solutions of the single quasilinear Schrödinger equation corresponding to system (1.3) by the same methods and obtained similar results. In particular, by the perturbation method, Liu et al.[9] considered the existence and multiplicity of solutions for the following quasilinear equation of the form

(1.4)

under suitable assumptions.

It is worth pointing out that the existence of one-bump or multi-bump bound state solutions for the related semilinear Schrödinger equation (1.2) for has been extensively studied. One can see Bartsch and Wang [10], Ambrosetti et al.[11], Ambrosetti et al.[12], Byeon and Wang [13], Cingolani and Lazzo [14], Cingolani and Nolasco [15], Del Pino and Felmer [16,17], Floer and Weinstein [18], Oh [19,20] and the references therein.

Motivated by the single equation (1.4), the purpose of this paper is to study the existence of both positive and negative solutions for the coupled quasilinear system (1.1). We mainly follow the idea of Liu et al.[9] to perturb the functional and obtain our main results. We point out that the procedure to system (1.1) is not trivial at all. Since the appearance of the quasilinear terms and , we need more delicate estimates.

The paper is organized as follows. In Section 2, we introduce a perturbation of the functional and give our main results (Theorem 2.1 and Theorem 2.2). In Section 3, we verify the Palais-Smale condition for the perturbed functional. Section 4 is devoted to some asymptotic behavior of the sequences and satisfying some conditions. Finally, our main results will be proved in Section 5.

Throughout this paper, we will use the same C to denote various generic positive constants, and we will use to denote quantities that tend to 0.

### 2 Perturbation of the functional and main results

In order to obtain the desired existence of solutions for system (1.1), in this section, we introduce a perturbation of the functional and give our main results.

The weak form of system (1.1) is

(2.1)

for all , which is formally the variational formulation of the following functional:

(2.2)

We may define the derivative of at in the direction of as follows:

(2.3)

We call a critical point of if , , and for all . That is, is a weak solution for system (1.1).

When we consider system (1.1) by using the classical critical point theory, we encounter the difficulties due to the lack of an appropriate working space. In general, it seems that there is no suitable space in which the variational functional possesses both smoothness and compactness properties. For smoothness, one would need to work in a space smaller than to control the term involving the quasilinear term in system (1.1), but it seems impossible to obtain bounds for sequence in this setting. Several ideas and approaches, such as minimizations [1,21], the Nehari method [5] and change of variables [2,3], have been used in recent years to overcome the difficulties. In this paper, we consider the perturbed functional

(2.4)

where is a parameter. Then it is easy to see that is a -functional on . We can define also the derivative of at in the direction of as follows:

(2.5)

for all . The idea of this paper is to obtain the existence of the critical points of for small and establish suitable estimates for the critical points as so that we may pass to the limit to get the solutions for the original system (1.1).

Our main results are as follows.

Theorem 2.1Assume that (A1)-(A3) hold, , and. Letand letbe a sequence of critical points ofsatisfyingandfor someCindependent ofn. Then, up to a subsequence,

as, andis a critical point of.

Theorem 2.2Assume that (A1)-(A3) hold, , and. Thenhas a positive critical pointand a negative critical point, and (resp.,) converges to a positive (resp., negative) solution for system (1.1) as.

Notation We denote by the norm of and by the norm of ().

### 3 Compactness of the perturbed functional

In this section, we verify the Palais-Smale condition ( condition in short) for the perturbed functional . We have the following proposition.

Proposition 3.1Forfixed, the functionalsatisfiescondition for all. That is, any sequencesatisfying, for,

(3.1)

(3.2)

has a strongly convergent subsequence in, whereis the dual space of.

To give the proof of Proposition 3.1, we need the following lemma firstly.

Lemma 3.2Suppose that a sequencesatisfies (3.1) and (3.2). Then

Proof It follows from (3.1) and (3.2) that

Thus we have

This completes the proof of Lemma 3.2. □

Now we give the proof of Proposition 3.1.

Proof of Proposition 3.1 From Lemma 3.2 , we know that is bounded in . So there exists a subsequence of , still denoted by , such that

Now we prove that in . In (2.5), choosing , we have

(3.3)

We may estimate the terms involved as follows:

Returning to (3.3), we have

which implies that , i.e., in . This completes the proof of Proposition 3.1. □

### 4 Some asymptotic behavior

Proposition 3.1 enables us to apply minimax argument to the functional . In this section, we also study the behavior of the sequences and satisfying

(4.1)

(4.2)

(4.3)

The following proposition is the key of this section.

Proposition 4.1Assume that the sequencesandsatisfy (4.1)-(4.3). Then, after extracting a sequence, still denoted byn, we have

and

as.

Proof Similar to the proof of Lemma 3.2, by (4.1)-(4.3), we have

(4.4)

Thus

(4.5)

for some C independent of n. Then, up to a subsequence, we have

and

as . This completes the proof of Proposition 4.1. □

### 5 Proof of main results

In this section, we give the proof of our main results. Firstly, we prove Theorem 2.1.

Proof of Theorem 2.1 Note that satisfies the following equation:

(5.1)

for all . Since

and

By Moser’s iteration, we have

(5.2)

Hence

(5.3)

for some C independent of n. To show that is a critical point of , we use some arguments in [22,23] (see more references therein). In (5.1), we choose , , where , , , and is a constant. Substituting into (5.1), we have

(5.4)

Note that , are positive for M large enough. By Fatou’s lemma, the weak convergence of and the fact that is bounded, we have

(5.5)

Let , . We may choose , such that , and . Then we obtain

(5.6)

for all , .

Similarly, we may obtain an opposite inequality. Thus we have

(5.7)

for all . That is, is a critical point of and a solution for system (1.1). By doing approximations, we have in the place of of (5.7)

(5.8)

Setting in (5.1), we have

(5.9)

Using as , (5.8), (5.9) and lower semi-continuity, we obtain

as .

In particular, we have

and

as . This completes the proof of Theorem 2.1. □

Next, we apply the mountain pass theorem to obtain the existence of critical points of . Set

for .

Let us consider the functional

(5.10)

Here and in what follows, we denote . The functional satisfies condition. Similarly, we may verify that satisfies condition. By the ε-Young inequality, for any , there exists such that

and

Then

for ε, ρ small. Thus we have

for and for small enough. Choose , and . Define a path by . When T is large enough, we have

and

for some m independent of .

Define

where

From the mountain pass theorem we obtain that

is a critical value of .

Let be a critical point corresponding to . We have . Thus is a positive critical point of by the strong maximum principle. In summary, we have the following.

Proposition 5.1There exist positive constantsρandmindependent ofμsuch thathas a positive critical pointsatisfying

Finally, we give the proof of Theorem 2.2.

Proof of Theorem 2.2 For a positive solution of system (1.1), the proof follows from Proposition 5.1 and Theorem 2.1. A similar argument gives a negative solution of system (1.1). This completes the proof of Theorem 2.2. □

### Competing interests

The authors declare that they have no competing interests.

### Authors’ contributions

All the authors were involved in carrying out this study. All authors read and approved the final manuscript.

### Acknowledgements

This paper was finished while the first author was a visiting fellow at the School of Mathematical Sciences of Beijing Normal University, and the first author would like to express her gratitude for their hospitality during her visit. This work is supported by the National Science Foundation of China (11061031), Fundamental Research Funds for the Central Universities (31920130004) and Fundamental Research Funds for the Gansu University.

### References

1. Poppenberg, M, Schmitt, K, Wang, Z: On the existence of soliton solutions to quasilinear Schrödinger equations. Calc. Var. Partial Differ. Equ.. 14, 329–344 (2002). Publisher Full Text

2. Liu, J, Wang, Y, Wang, Z: Soliton solutions for quasilinear Schrödinger equation, II. J. Differ. Equ.. 187, 473–493 (2003). Publisher Full Text

3. Colin, M, Jeanjean, L: Solutions for a quasilinear Schrödinger equation: a dual approach. Nonlinear Anal.. 56, 213–226 (2004). Publisher Full Text

4. Berestycki, H, Lions, PL: Nonlinear scalar field equations, I. Arch. Ration. Mech. Anal.. 82, 313–346 (1983)

5. Liu, J, Wang, Y, Wang, Z: Solutions for quasilinear Schrödinger equations via the Nehari method. Commun. Partial Differ. Equ.. 29, 879–901 (2004). Publisher Full Text

6. Lions, PL: The concentration-compactness principle in the calculus of variations. The locally compact case. Part I. Ann. Inst. Henri Poincaré, Anal. Non Linéaire. 1, 109–145 (1984)

7. Guo, Y, Tang, Z: Ground state solutions for the quasilinear Schrödinger systems. J. Math. Anal. Appl.. 389, 322–339 (2012). Publisher Full Text

8. Guo, Y, Tang, Z: Ground state solutions for the quasilinear Schrödinger equation. Nonlinear Anal.. 75, 3235–3248 (2012). Publisher Full Text

9. Liu, X, Liu, J, Wang, Z: Quasilinear elliptic equations via perturbation method. Proc. Am. Math. Soc.. 141, 253–263 (2013)

10. Bartsch, T, Wang, Z: Multiple positive solutions for a nonlinear Schrödinger equation. Z. Angew. Math. Phys.. 51, 366–384 (2000)

11. Ambrosetti, A, Badiale, M, Cingolani, S: Semiclassical states of nonlinear Schrödinger equations. Arch. Ration. Mech. Anal.. 140, 285–300 (1997). Publisher Full Text

12. Ambrosetti, A, Malchiodi, A, Secchi, S: Multiplicity results for some nonlinear Schrödinger equations with potentials. Arch. Ration. Mech. Anal.. 159, 253–271 (2001). Publisher Full Text

13. Byeon, J, Wang, Z: Standing waves with a critical frequency for nonlinear Schrödinger equations, II. Calc. Var. Partial Differ. Equ.. 18, 207–219 (2003). Publisher Full Text

14. Cingolani, S, Lazzo, M: Multiple positive solutions to nonlinear Schrödinger equations with competing potential functions. J. Differ. Equ.. 160, 118–138 (2000). Publisher Full Text

15. Cingolani, S, Nolasco, M: Multi-peaks periodic semiclassical states for a class of nonlinear Schrödinger equations. Proc. R. Soc. Edinb.. 128, 1249–1260 (1998). Publisher Full Text

16. Del Pino, M, Felmer, P: Semi-classical states for nonlinear Schrödinger equations. Ann. Inst. Henri Poincaré. 15, 127–149 (1998). Publisher Full Text

17. Del Pino, M, Felmer, P: Multi-peak bound states for nonlinear Schrödinger equations. J. Funct. Anal.. 149, 245–265 (1997). Publisher Full Text

18. Floer, A, Weinstein, A: Nonspreading wave packets for the cubic Schrödinger equation with a bounded potential. J. Funct. Anal.. 69, 397–408 (1986). Publisher Full Text

19. Oh, YG: On positive multi-bump bound states of nonlinear Schrödinger equations under multiple well potential. Commun. Math. Phys.. 131, 223–253 (1990). Publisher Full Text

20. Oh, YG: Existence of semiclassical bound states of nonlinear Schrödinger equations with potentials of class . Commun. Partial Differ. Equ.. 13, 1499–1519 (1988). Publisher Full Text

21. Liu, J, Wang, Z: Soliton solutions for quasilinear Schrödinger equations, I. Proc. Am. Math. Soc.. 131, 441–448 (2003). Publisher Full Text

22. Canino, A, Degiovanni, M: Nonsmooth critical point theory and quasilinear elliptic equations. (1995)

23. Liu, J, Wang, Z: Bifurcations for quasilinear Schrödinger equations, II. Commun. Contemp. Math.. 10, 723–743 (2008)

format: plain


text:
# Is this a triangle?

Write a function/program that, given three positive integers a, b and c, prints a Truthy value if a triangle (any triangle) could have side lengths a, b and c and outputs a Falsy value otherwise.

# Input

Three positive integers in any sensible format, for example:

• three distinct function arguments, f(2, 3, 5)
• a list of three numbers, [2,3,5]
• a comma-separated string, "2,3,5"

You may not assume the inputs are sorted.

# Output

A Truthy value if a triangle could have the sides with the lengths given, Falsy otherwise.

# Test cases

1, 1, 1 -> Truthy
1, 2, 3 -> Falsy
2, 1, 3 -> Falsy
1, 3, 2 -> Falsy
3, 2, 1 -> Falsy
3, 1, 2 -> Falsy
2, 2, 2 -> Truthy
3, 4, 5 -> Truthy
3, 5, 4 -> Truthy
5, 3, 4 -> Truthy
5, 4, 3 -> Truthy
10, 9, 3 -> Truthy
100, 10, 10 -> Falsy

This is so shortest solution in bytes, wins. Consider upvoting this challenge if you have fun solving it and... Happy golfing!

• Related: count the number of triangles – Kevin Cruijssen Feb 12 '20 at 7:29
• You ask: "is this a triangle?" Probably not, but who knows. Is anything really a triangle? – Lyxal Feb 12 '20 at 9:47
• @Lyxal I say you can be anything. I believe in you. Do you? – RGS Feb 12 '20 at 13:14
• △ △ ▽ ▽ ◁ ▷ ◁ ▷ A B ␂ – S.S. Anne Feb 12 '20 at 15:19
• you should at least admit that your test-cases 1,2,3 are "borderline triangles" as the shorter sides summed give the longest side so you have a "triangle on a line" - while 100,10,10 definitely is false as 10+10 is shorter than 100 – eagle275 Feb 13 '20 at 7:33

# Python, 24 bytes

lambda l:sum(l)>max(l)*2

Try it online!

Checks if a+b+c > max(a,b,c)*2. If, say, c is the biggest one, this is equivalent to a+b+c>2*c, or a+b>c, which is want we want for the triangle inequality.

• This is also what I had for Python! +1 for us :p – RGS Feb 12 '20 at 7:22

# CP-1610 machine code (Intellivision), 12 DECLEs1 = 15 bytes

A routine taking $$\(a,b,c)\$$ into R0, R1 and R2 respectively and setting the carry if $$\(a,b,c)\$$ is not a triangle, or clearing it otherwise.

083     |         MOVR    R0,     R3
15A     |         CMPR    R3,     R2
049     |         SLL     R1
159     |         CMPR    R3,     R1
201 002 |         BC      @@rtn
048     |         SLL     R0
158     |         CMPR    R3,     R0
0AF     | @@rtn   JR      R5

### How?

$$\cases{a+b>c\\a+c>b\\b+c>a}$$

We test:

$$\cases{a+b>c\\a+b+c>2b\\a+b+c>2a}$$

The left parts of the inequalities are stored into R3. If the first test fails (when $$\a+b\$$ is stored into R3 and compared with R2), the carry is set and added to the program counter (R7), forcing the next instruction to be skipped. Consequently, R3 is not updated to $$\a+b+c\$$ and the last 2 comparisons are turned into:

$$\cases{a+b>2b\\a+b>2a}\Leftrightarrow\cases{a>b\\b>a}$$

which of course is never true, so the test is guaranteed to fail as expected.

All in all, this saves a branch which would have cost 1 extra DECLE.

### Full commented test code

        ROMW    10                ; use 10-bit ROM width
ORG     $4800 ; map this program at$4800

;; ------------------------------------------------------------- ;;
;;  main code                                                    ;;
;; ------------------------------------------------------------- ;;
main    PROC

SDBD                      ; set up an interrupt service routine
MVII    #isr,   R0        ; to do some minimal STIC initialization
MVO     R0,     $100 SWAP R0 MVO R0,$101

EIS                       ; enable interrupts

MVII    #$200, R4 ; R4 = pointer into backtab SDBD ; R5 = pointer into test cases MVII #tc, R5 MVII #13, R3 ; R3 = number of test cases @@loop MVI@ R5, R0 ; R0 = a MVI@ R5, R1 ; R1 = b MVI@ R5, R2 ; R2 = c PSHR R3 ; save R3 and R5 on the stack PSHR R5 CALL tr ; invoke our routine PULR R5 ; restore R3 and R5 PULR R3 MVII #$80,   R0        ; R0 = '0'
BC      @@draw

MVII    #$88, R0 ; or '1' if the carry is not set @@draw MVO@ R0, R4 ; draw this character DECR R3 ; next test case BNEQ @@loop DECR R7 ; loop forever ;; ------------------------------------------------------------- ;; ;; test cases ;; ;; ------------------------------------------------------------- ;; tc PROC DECLE 1, 1, 1 ; true DECLE 1, 2, 3 ; false DECLE 2, 1, 3 ; false DECLE 1, 3, 2 ; false DECLE 3, 2, 1 ; false DECLE 3, 1, 2 ; false DECLE 2, 2, 2 ; true DECLE 3, 4, 5 ; true DECLE 3, 5, 4 ; true DECLE 5, 3, 4 ; true DECLE 5, 4, 3 ; true DECLE 10, 9, 3 ; true DECLE 100, 10, 10 ; false ENDP ;; ------------------------------------------------------------- ;; ;; ISR ;; ;; ------------------------------------------------------------- ;; isr PROC MVO R0,$0020     ; enable display

CLRR    R0
MVO     R0,     $0030 ; no horizontal delay MVO R0,$0031     ; no vertical delay
MVO     R0,     $0032 ; no border extension MVII #$D,    R0
MVO     R0,     $0028 ; light-blue background MVO R0,$002C     ; light-blue border

JR      R5                ; return from ISR

ENDP

;; ------------------------------------------------------------- ;;
;;  our routine                                                  ;;
;; ------------------------------------------------------------- ;;
tr      PROC

MOVR    R0,     R3        ; R3 = a
ADDR    R1,     R3        ; R3 = a + b
CMPR    R3,     R2        ; is R3 greater than c?
ADCR    R7                ; if not, skip the next instruction

ADDR    R2,     R3        ; R3 = a + b + c (or still a + b if skipped)
SLL     R1                ; R1 = 2b
CMPR    R3,     R1        ; is R3 greater than 2b?
BC      @@rtn             ; if not, return with the carry set

SLL     R0                ; R0 = 2a
CMPR    R3,     R0        ; is R3 greater than 2a?
; if not, the carry is set

@@rtn   JR      R5                ; return

ENDP

### Output

screenshot from jzIntv

1. A CP-1610 opcode is encoded with a 10-bit value (0x000 to 0x3FF), known as a 'DECLE'.

• +1 Marvelous :D – RGS Feb 12 '20 at 9:21
• Do you have to run this on your TV? Maybe an emulator is in order... – S.S. Anne Feb 12 '20 at 15:15
• @S.S.Anne I'm doing 95% of the debugging on jzIntv. For real (much bigger) projects, I do upload the ROM to a real Intellivision from time to time. – Arnauld Feb 12 '20 at 15:26
• Oh, so that is an emulator. The homepage was unclear. – S.S. Anne Feb 12 '20 at 15:28
• I think this should state that it is in machine language, not assembly language. I see that the assembler mnemonics alone are typically four bytes per instruction. PS. +1 Very nice. – Bit Chaser Feb 12 '20 at 18:20

# Java 8, 524938 26 bytes

(a,b,c)->a+b>c&a+c>b&b+c>a

Port of @xnor's formula turns out to be shorter after all, by taking the input as three loose integers instead of a List/array.
-12 bytes thanks to @xnor for reminding me that the first 6-bytes formula I used in my 05AB1E answer is actually shorter in Java:

$$\(a+b>c)\land(a+c>b)\land(b+c>a)\$$

Try it online.

Explanation:

(a,b,c)->  // Method with three integer parameters and boolean return-type
a+b>c    //  Return whether the sum of a and b is larger than c
&a+c>b   //  and the sum of a and c is larger than b
&b+c>a   //  and the sum of b and c is larger than a

S->{S.sort(null);return S.pop()<S.pop()+S.pop();}

Port of @GB's Ruby answer with input as a Stack of Integers.

Try it online.

Explanation:

S->{                        // Method with Integer-Stack parameter and boolean return-type
S.sort(null);             //  Sort the input-Stack
return S.pop()            //  Check if the last largest value
<S.pop()+S.pop();} //  is smaller than the lowest two values added together

• It is funny that even though you compute the max with a bunch of ?: operators, you can still get it to be shorter than the other answer! – RGS Feb 12 '20 at 13:20
• Wow! Just looking over the other answers and see we're letter for letter identical (save the C/Java diffs)! I swear I did mine on my own! T_T – Noodle9 Feb 12 '20 at 13:20
• @RGS Yeah, and the ?: are actually shorter than using the Math.max builtin, since it can only take 2 arguments (Math.max(Math.max(a,b),Math.max(b,c)) is longer than (a>b?a>c?a:c:b>c?b:c)). PS: as List with builtin streams it's even longer: L->L.stream().mapToInt(i->i).sum()>L.stream().max((i,j)->i.compareTo(j)).get()*2 (80 bytes). That was my very first attempt, before I posted my answer and had the 51 bytes alternative. ;) – Kevin Cruijssen Feb 12 '20 at 13:27
• @RGS That's why it's so fun and I'm still here after almost four years of code-golfing. Although Java will never win a code-golf contest, tbh. Unless it's the only answer of course. ;P – Kevin Cruijssen Feb 12 '20 at 13:36
• It looks like it's actually shorter to write out a+b>c&b+c>a&c+a>b: TIO – xnor Feb 12 '20 at 23:53

# Ruby, 19 bytes

->*a{2*a.max<a.sum}

You can try it online!

Uses the fact that this Ruby answer said it didn't want to implement the port of xnor's answer and at the same time taught me enough syntax to guess how the max of an array is calculated :)

• @S.S.Anne Please remember to follow the Code of Conduct. – Doorknob Feb 15 '20 at 4:45
• @Doorknob What did I do? – S.S. Anne Feb 15 '20 at 14:47

# Ruby, 23 bytes

->*a{a.sort!.pop<a.sum}

Try it online!

A different approach, xnor's formula would be shorter but I'm satisfied with that.

• +1, I'll take it then :) – RGS Feb 12 '20 at 8:53

fn($a)=>max($a)*2<array_sum($a);  Try it online! Lambda function that takes an array [a, b, c] and outputs empty string if false or "1" if true, with xnor's formula. Note that in PHP the ; is only necessary when the function is attributed to a variable, so it's placed in the footer (provided code is valid as it is). EDIT: thanks to Guillermo Phillips for introducing me to PHP 7.4 short notation! (as a declaration without {}, it now needs the ;) • Could you not use the short lambda syntax from PHP 7.4? – Guillermo Phillips Feb 12 '20 at 10:25 • @GuillermoPhillips thanks a lot, I actually didn't know them, it doesn't appear on the manual (yet?) – Kaddath Feb 12 '20 at 10:32 • Good job on this PHP answer and +1 for this lambda function! – RGS Feb 12 '20 at 13:15 # Wolfram Language (Mathematica), 12 bytes 2Max@#<Tr@#&  Try it online! • Thanks for teaching me smth! I'm assuming (1): the single @ can be used to call a function and (2): the trace of a list is its sum. Am I right? – RGS Feb 12 '20 at 13:31 • @RGS Yup! and here are some "master tips"... codegolf.stackexchange.com/q/12900/67961 – J42161217 Feb 12 '20 at 14:32 # 05AB1E, 65 4 bytes O;‹ß  -1 byte by porting @xnor's algorithm. -1 byte thanks to @Grimmy using a derived formula. Explanation: The formula this program uses to determine if three side-lengths $$\a,b,c\$$ form a triangle is: $$\t=\frac{a+b+c}{2}\$$ $$\(a O # Take the sum of the (implicit) input-list ; # Halve it ‹ # Check if it's larger than each of the values of the (implicit) input-list ß # Get the minimum of those checks # (P can be used as alternative to check if all are truthy instead) # (after which this result is output implicitly)  Original 6-byter: ÀĆü+‹P  Explanation: The formula this program uses to determine if three side-lengths $$\a,b,c\$$ form a triangle is: $$\a+b>c\$$ $$\a+c>b\$$ $$\b+c>a\$$ Which translates to the following code for 05AB1E: À # Rotate the (implicit) input-list once towards the right: [a,b,c] → [b,c,a] Ć # Enclose it; appending its head to itself: [b,c,a] → [b,c,a,b] ü+ # Sum each overlapping pair: [b,c,a,b] → [b+c,c+a,a+b] ‹ # Check for each whether it's larger than the (implicit) input-list: # [a<b+c,b<c+a,c<a+b] P # And check if all three are truthy by taking the product # (ß could be used as alternative, like in the 4-byter program above) # (after which the result is output implicitly)  # C (gcc), 30 bytes Probably the most basic formula. f(a,b,c){a=a+b>c&a+c>b&b+c>a;}  Try it online! • Good job recognising the more verbose formula ends up being shorter here! What is the leading a=? Is it for some kind of implicit return? – RGS Feb 12 '20 at 18:39 • @RGS Yes. The way gcc compiles this, assigning the value to a variable or returning it both put the result in eax and therefore lead to the same behavior. – Arnauld Feb 12 '20 at 19:16 • I learned so much of C from this answer. I didn't know you could omit the type of parameters, return type, and return statement. – user1754322 Feb 15 '20 at 11:38 • @user1754322 Just remember to only use it in code golf. This is formally undefined behavior in every C standard. – S.S. Anne Feb 15 '20 at 14:49 # GolfScript, 5 bytes $~\->

Why bother with any of the other math? If the longest side is longer (or equal) than the sum of the shorter two, it can't be a triangle. Otherwise it can.

Input is an array. Output is 0 or 1. If you have the list necessarily sorted, then you can just input as a dumped array (straight onto the stack) and the first two characters are unneeded. If the list can be entered in reverse order, then you can remove the first three characters and flip the greater-than to a lesser-than, leaving only 2 characters needed.

$~\-> #Check if side lengths could make a triangle$      #Sort
~     #Dump array onto stack (stack is now 1 2 3)
\    #Swap the top two elements (1 3 2)
-   #Subtract the top from the second (1 3-2)
>  #Check if that difference is strictly greater than the smallest value

This is equivalent to my last answer, but instead of a+b>c, I did a>c-b, which saves a character of stack-shuffling.

Try it online!

• you are still using maths :D good job on this! – RGS Feb 12 '20 at 13:32

# APL (Dyalog Unicode), 7 bytesSBCS

+/>2×⌈/

Try it online!

Also uses xnor's formula of sum(a,b,c) > 2 * max(a,b,c).

### How it works

+/>2×⌈/
+/       ⍝ Is sum
>      ⍝ greater than
2×    ⍝ twice of
⌈/  ⍝ max?

# Alternative 7 bytesSBCS

∧/+/>+⍨

Try it online!

### How it works

∧/+/>+⍨
+/     ⍝ Is sum
>    ⍝ greater than
+⍨  ⍝ twice each number?
∧/       ⍝ All of them?

• Thanks for the submission +1 what is up with the weird character you are using in the explanations? – RGS Feb 12 '20 at 8:11
• @RGS If you mean ⍝, it starts a line comment in APL (just like // in C and # in Python). – Bubbler Feb 12 '20 at 8:15
• @Bubbler But only C99 and after. It's commonly called a "C++-style comment". – S.S. Anne Feb 12 '20 at 15:16

# Jelly, 4 bytes

ṀḤ<S

Try it online!

A monadic link taking a list of integers and returning a Jelly boolean (1 = True, 0 = False).

Based on @xnor's Python answer so be sure to upvote that one too!

## Explanation

Ṁ    | Maximum
Ḥ   | Doubled
<  | Less than:
S | - Sum of original argument

• +1 good job on the port :) – RGS Feb 12 '20 at 8:48
• @RGS thanks. Nice series of questions recently! – Nick Kennedy Feb 12 '20 at 8:51
• thanks for your feedback, let's keep them coming! – RGS Feb 12 '20 at 8:57
• Alternative 4 bytes: Ḥ<SẠ for "double of all elements less than sum". – Bubbler Feb 12 '20 at 9:05

# Retina 0.8.2, 31 bytes

\d+
$* O1+ ^(?!(1+),(1+),\1\2)  Try it online! Link includes test cases. Explanation: \d+$*

Convert to unary.

O1+

Sort.

^(?!(1+),(1+),\1\2)

Check that the sum of the first two is greater than the third.

• These Retina solutions keep amusing me! +1 – RGS Feb 12 '20 at 13:17
• @RGS It's always nice when you can coax Retina to perform functions which seem unrelated to string manipulation. – Neil Feb 13 '20 at 10:06
• Indeed!!! Now I want to see how Retina fares in the count regex matches challenge, which is related to string manipulation! – RGS Feb 13 '20 at 11:28
• @RGS I can't see an obvious approach for that one sorry. – Neil Feb 13 '20 at 18:47

# C (gcc), 54 $$\\cdots\$$ 42 36 bytes

Saved 6 bytes thanks to ceilingcat!!!

f(a,b,c){a=a+b+c>2*fmax(a>b?a:b,c);}

Try it online!

Uses xnor's formula.

• Obvious +1 for having the same answer. ;) – Kevin Cruijssen Feb 12 '20 at 13:23
• @KevinCruijssen Likewise my good man. :-) – Noodle9 Feb 12 '20 at 13:33
• You can use the same golf as my Java answer (credit to @xnor): f(a,b,c){a=a+b>c&a+c>b&b+c>a;} EDIT: Nvm, @Arnauld already posted that as a separated C answer. – Kevin Cruijssen Feb 13 '20 at 7:43
• @KevinCruijssen Ah, interesting! Ok, I'll stick with what I've got - thanks anyway! :-) – Noodle9 Feb 13 '20 at 9:49
• @ceilingcat Nice one - thanks! :-) – Noodle9 Jul 24 '20 at 0:12

# Burlesque, 13 bytes

raJ++j>]2.*.>

Try it online!

ra # Read as array
J  # Duplicate
++ # Sum
j  # Swap
>] # Maximum
2.*# Double
.> # Greater than

• +1 good job working on this :) – RGS Feb 12 '20 at 8:58

# Wren, 54 bytes

I haven't written Wren in a long time...

Fn.new{|x|x.reduce{|a,b|a+b}>x.reduce{|a,b|a>b?a:b}*2}

Try it online!

• I have never seen Wren around here, for the short time I've been around – RGS Feb 12 '20 at 13:25

# C++ (gcc), 89 bytes

Array/vector solution as shown in other answers.

#import<regex>
int f(std::vector<int>v){std::sort(&v[0],&*end(v));return v[2]<v[1]+v[0];}

Try it online!

# C++ (gcc), 56 bytes

int f(int a,int b,int c){a=a+b+c>2*((a=a>b?a:b)>c?a:c);}

Like the C solution but with more boilerplate.

-2 bytes thanks (indirectly) to Arnauld!

Try it online!

• Good job on this one! +1 can you please explain me why you need the #import? – RGS Feb 12 '20 at 15:56
• @RGS Because functions (like std::sort) are not implicitly declared in C++, and std::vector is not a builtin. – S.S. Anne Feb 12 '20 at 16:22

# C (gcc), 40 bytes

-2 bytes thanks to Arnauld: stores the result of max in a. Eliminates the variable m used in the previous solution.

f(a,b,c){a=a+b+c>2*((a=a>b?a:b)>c?a:c);}

Try it online!

# C (gcc), 42 bytes

f(a,b,c){a=a+b+c>2*(a>b?a>c?a:c:b>c?b:c);}

Try it online!

• Good solution +1! It is a shame that the m doesn't save you anything... – RGS Feb 12 '20 at 15:06

Never done one of these before. Here are two algorithms borrowed from other answers implemented in JavaScript.

# JavaScript, 32 bytes

(a,b,c)=>a+b+c>Math.max(a,b,c)*2

Try it online

# JavaScript, 28 bytes

(a,b,c)=>a=a+b>c&a+c>b&b+c>a

Try it online

• Really good job James! +1 And welcome to the site :D – RGS Feb 13 '20 at 11:48

# Lua, 128 120 bytes

b=io.read()t={}for r in b.gmatch(b,"([^,]+)")do
table.insert(t,r) end
print(t[1]+t[2]+t[3]+0>math.max(t[1],t[2],t[3])*2)

Try it online!

# Lua, 37 bytes

@Jo King's solution using TIO properly and following the rules of the challenge.

a,b,c=...print(a+b+c>math.max(...)*2)

Try it online!

Very direct translation of @xnor's algorithm

• Hey there, if I understand correctly, you are spending quite some bytes reading from input and extracting the integers. Can't you define a shorter function that already takes the integers? And then you do the IO parsing in the header section of TIO? – RGS Feb 12 '20 at 8:48
• @RGS, Yes, absolutely. I can post a solution like that in the comments. Just gimmee a bit.... – ouflak Feb 12 '20 at 8:49
• no need to be in the comments! Rewrite your answer. In this community, and for code-golf, you can usually just write the function that does what is asked, as long as your function accepts the input as specified by the OP. I can't really elaborate on it now nor exemplify with Lua because I don't know lua, but I'm sure you can reduce your byte count by a lot if you do what I mean – RGS Feb 12 '20 at 8:57
• @JoKing, I've really got do some study and wrap my head around how to use TIO's header properly. – ouflak Feb 13 '20 at 11:51

# Pepe, 93 bytes

REeEREeEREeEREEEEeEeEREEEeREEEEEEEREEEEEEEErREEEEEerEEEEEerEeEeErEEEEeeRErREErEEEEeREeReEreEE

Input: a;b;c
Output: -1 for falsy, 1 for truthy

Explanation:

REeEREeEREeE # Push 3 inputs (num) -> (R)
REEEEeEeE # Sort the (R) stack
REEEe # Move pointer pos to the last -> (R)
REEEEEEE # Move last item of (R) (num c in this case) to (r)
REEEEEEEE # Sum the stack (note: does not remove the stack completely!) -> (R)
# For some reason, the pointer pos of (R) keeps sticking to the last item
rREEEEEe # Subtract active item of (R) (a+b) to (r) (num c) -> (R)
# ((a+b)-c)
# r flag: Preserve the items
rEEEEEe # Same as above, but put it to (r)
# This switches the expression: (c-(a+b)) or -((a+b)-c)
# No more r flag this time, so remove these two items
rEeEeE # Absolute of (r)
rEEEEee # Divide active item of (R) (a+b) and (r) (num c) -> (r)
RE # Push 0 -> (R)
rREE # Create loop labelled 0 -> (R)
# r flag: Skip until Ee (or REe)
rEEEEe # Decrement (0 -> -1) -> (r)
ReE # If the division of (r) returned 0, go inside the loop
reEE # Output (r) as number

Try it online!

• Really funny solution! +1 good job! – RGS Feb 15 '20 at 8:30

# x86-16 machine code, 21 bytes

8B D0       MOV  DX, AX     ; DX = a
03 C3       ADD  AX, BX     ; AX = a + b
3B C1       CMP  AX, CX     ; is a + b > c?
76 0C       JBE  NOT_TRI    ; if so, not triangle
03 C1       ADD  AX, CX     ; AX = a + b + c
D1 E2       SHL  DX, 1      ; DX = 2a
3B C2       CMP  AX, DX     ; is a + b + c > 2a?
76 04       JBE  NOT_TRI    ; if so, not triangle
D1 E3       SHL  BX, 1      ; BX = 2b
3B C3       CMP  AX, BX     ; is a + b + c > 2b?
NOT_TRI:

Callable function, input AX, BX and CX. Result is in ZF: NZ if triangle, ZR if not.

Uses Arnauld's method to check.

I/O from DOS test program:

For some reason the test program returns "Hotdog" if a triangle and "Not Hotdog" if not.

• Nice submission! Are you being serious with the "for some reason" or did you make the program return (not) hot dog for Truthy and Falsy? :) – RGS Feb 15 '20 at 21:50
• @RGS okay, you got me, maybe it was kinda intentional. – 640KB Feb 16 '20 at 14:49

# MathGolf, 5 bytes

Σ\╙∞>

Try it online.

Exact port, including same explanation, as my 5-byte 05AB1E answer: sum; swap; max; double; a>b.

• But in this case, it is shorter! – RGS Feb 12 '20 at 8:12
• @RGS What do you mean? Both are 5 bytes, and the exact same explanation. – Kevin Cruijssen Feb 12 '20 at 8:14
• I meant your original O5AB1E one was 6 bytes :p – RGS Feb 12 '20 at 8:45

# J, 8 bytes

+/>2*>./

Try it online!

Uses xnor's formula

• +1 simple port, right? :) – RGS Feb 12 '20 at 8:12
• @RGS Yes, it's almost identical to Bubbler's APL solution - it's normal, J is APL's relative. – Galen Ivanov Feb 12 '20 at 8:30

# GolfScript, 14 bytes

A port of the Ruby answers.

~.{+}*$)\;2*>  Try it online! ## Explanation ~ # Evaluate the input . # Copy it twice {+}* # Sum the input$       # Sort the other input
)\;    # Select the last item
# in the sorted list
2*  # Double this item
> # Compare

• Good job on this answer! – RGS Feb 12 '20 at 13:22

## Wd, 12 10 bytes

I'm quite satisfied because W doesn't have a max function. OR a function sorting the array.

▼╪m╜w♣S×∟╖

## Uncompressed:

<a&b|R       % Max.
2*     % Double.
S    % Swap.
+r  % Sum.
< % Less than.
% It's in the wrong order because
% I can golf a few bytes off the max function.

% BTW the max function is implemented like this:
(b<a) && (a) || (b)
% The reduction reduces this function over the whole list.

• Impressive first answer and very well presented. welcome! – J42161217 Feb 12 '20 at 12:35
• @J42161217 ditto! – RGS Feb 12 '20 at 13:27
• I feel like there was another higher-up user named a'_' that went missing recently. Hmm... – S.S. Anne Feb 12 '20 at 14:53

# Keg, -hr, 7 5 bytes

÷⑭$->  Try it online! Woot! Port of the 5-byte golfscript answer. TIO won't work because the version stored has a bug with the ⑭ command, while the most recent version doesn't. (For those interested, I believe TIO is ~20 commits behind the official repo). This gets converted into the following python program: from KegLib import * from Stackd import Stack stack = Stack() printed = False item_split(stack) sort_stack(stack) swap(stack) maths(stack, '-') comparative(stack, '>') if not printed: raw(stack)  ## Explained ÷  First, we take the sides as input. Keg doesn't do lists very well at the moment (I'm working on improving that though), so each number needs to be taken individually. The ¿ takes the input and evaluates it as a literal. Forget I ever said that. One can actually take a list as input lol. So instead of taking three individual numbers, we go ahead and item split the implicit input list. ⑭$->

Then, like the golfscript answer, we sort the stack (⑭), swap the top two items (\$) subtract those two items (-) and then finally compare the two with >.

• Kool answer! +1 To bad you have to spend three bytes reading the integers! – RGS Feb 12 '20 at 23:11
• @RGS whatever do you mean by spending three bytes to read the integers? :P – Lyxal Feb 12 '20 at 23:14
• Nevermind me :) Can't upvote anymore today :( will try to remember to do it tomorrow! – RGS Feb 12 '20 at 23:19
• @RGS I know that feeling :P. – Lyxal Feb 12 '20 at 23:20
• one up!! Go work on those lists now!!! – RGS Feb 13 '20 at 0:41

# Stax, 7 bytes

å·b→1.R

Run and debug it

• Haven't heard of Stax before! Good work on this +1 – RGS Feb 13 '20 at 8:36

# Excel, 20 bytes

=SUM(A:A)>MAX(A:A)*2

Input in A1, A2 and A3.

Alternatively, with input on A1, B1 and C1:

=SUM(1:1)>MAX(1:1)*2

• Cool submission! :D +1 – RGS Feb 14 '20 at 12:46
• Interestingly, SUM(A:A) is exactly the same number of bytes as A1+A2+A3 – Calculuswhiz Jul 25 '20 at 3:05

# Julia 1.0, 20 bytes

f(s)=all(sum(s).>2s)

Try it online!

• Nice and simple Julia submission +1 :) – RGS Feb 17 '20 at 8:03


text: Ohm’s Law and Joule’s Law are commonly used in calculations dealing with electronic circuits. 2. REFERENCES : 1. Electrical Formulas helps us to calculate the parameters related to electricals in any electrical components.. the following formulas: E = I x R I = E / R R = E / I P = E x I Direct Current Electricity An electrical current can flow in either of two directions through a conductor. Performance & security by Cloudflare, Please complete the security check to access. Electrical & Electronics Engineering Basic Formulas. This article deals with electrical formulas and their derivations. The following is a brief description of the information presented in each module of the handbook. Your IP: 185.56.147.7 1. Note: Active power is the "resistive" … Completing the CAPTCHA proves you are a human and gives you temporary access to the web property. With just a handful of basic mathematical formulas, you can get pretty far in analyzing the goings-on in electronic circuits and in choosing values for electronic components in circuits you design. Solved Examples. For the circuit shown, Kirchhoff’s Laws tells you the following: KVL: Vbattery – VR – VLED = 0, or Vbattery = VR + VLED. Voltage - the electrical pressure needed to force current through any load. times the peak value. If you are at an office or shared network, you can ask the network administrator to run a scan across the network looking for misconfigured or infected devices. In these calculations: Electronic circuits may contain resistors or capacitors in series, parallel, or a combination. 3. Circuit designers use RC networks to produce simple timers and oscillators because the charge time is predictable and depends on the values of the resistor and the capacitor. The formula for a average voltage is Vav =0.636Vmax (12.5) Where Vav is the average voltage for one alteration, and Vmax is the maximum or peak voltage. Furthermore, there exist four basic physical quantities of electricity. First of all, understanding the basics of electricity is vital for gaining proper knowledge of electricity. So if you know two of the three characteristics, your can calculate the third one. Another way to prevent getting this page in the future is to use Privacy Pass. Electrical formulae for current, voltage, resistance, power and conductivity.The units and formula for various electrical parameters included in the page. o 40293 10895 PRINTED IN U.S.A. A Division of Tandy Corporation Fort Worth, TX 76102 Rad.e 'haek . Units - Volts, V: Measured line to line or line to neutral with a voltmeter. The complete list of electrical & electronics engineering basic formulas cheat sheet for PDF download to help users to use them offline to learn or workout how to execute or solve the various calculations of voltage, current, resistance, conductance, capacitance, inductance, impedance, resistor color coding, voltage divider, AC induction motor characteristics, motor starting & running current, induction motor slip speed… You may need to download version 2.0 now from the Chrome Web Store. Equivalent Resistance - Series & Parallel Circuit, Equivalent Capacitance - Series & Parallel Circuit, Equivalent Inductance - Series & Parallel Circuit, Equivalent Impedance - RLC Series Circuit, Equivalent Impedance - RLC Parallel Circuit, 3 Phase AC Motor Starting & Running Current. Cloudflare Ray ID: 5f7d6a10391f0c21 You can determine the equivalent value of resistance or capacitance using the following formulas: Kirchhoff’s Circuit Laws are commonly used to analyze what’s going on in a closed loop circuit. The foundation is a part of the building that isn’t visible in the finished structure, but is essential in erecting a building that will have the necessary strength to endure. Ohm's Law Formulas for A-C Circuits and Power Factor. Ohm’s Law and Joule’s Law are commonly used in calculations dealing with electronic circuits. Academia.edu is a platform for academics to share research papers. These laws are straightforward, but when you’re trying to solve for one variable or another, it is easy to get them confused. Almost all the projects in class will be powered by DC electricity. What is the monthly payment for $200000 over 5 years. The Electrical Science handbook consists of fifteen modules that are contained in four volumes. Displaying Electrical Signals on an Oscilloscope, Part of Electronics For Dummies Cheat Sheet. These laws are straightforward, but when you’re trying to solve for one variable or another, it is easy to get them confused. Sukhija Oxford University Press. In these calculations:V = voltage (in volts)I = current (in amps)R = resistance (in ohms)P = power (in watts) First of all, understanding the basics of electricity is vital for gaining proper knowledge of electricity. The complete list of electrical & electronics engineering basic formulas cheat sheet for PDF download to help users to use them offline to learn or workout how to execute or solve the various calculations of voltage, current, resistance, conductance, capacitance, inductance, impedance, resistor color coding, voltage divider, AC induction motor characteristics, motor starting & running current, induction motor slip speed, energy & charge in capacitor, quality factor, single & three phase AC transmission lines, DC transmission lines or any closed or open circuits characteristics. Electrical formulae for current, voltage, resistance, power and conductivity.The units and formula for various electrical parameters included in the page. INTRODUCTION TO UNIT 1—ELECTRICIAN’S MATH AND BASIC ELECTRICAL FORMULAS. In the above formulas 1 is the angle of lead or lag between current and voltage and cos 1 = P/EI = power factor or pf. If it flows in only one direction whether steadily or in pulses, it is called direct current (DC). Nominal System Voltage Actual System Voltage (use for calculations) Single Phase Similarly, the formula for average current is Iav =0.636Imax (12.6) Where Iav is the average current for one alteration, and Imax is the maximum or peak current. Furthermore, there exist four basic physical quantities of electricity. Basic Electrical Engineering – By M.S.Naidu and S. Kamakshiah TMH. Basic Electrical Engineering By T.K.Nagasarkar and M.S. After the equivalent of one time constant has passed, a discharged capacitor will charge to roughly two-thirds its capacity, and a charged capacitor will discharge nearly two-thirds of the way. Ohm’s Law is the basic formula used in all AC and DC electrical circuits. In order to construct a building that will last into the future, a strong foundation is a prerequisite. • Solution: Given: Current I = 4 A, Resistance R = 5$\omega$Example 1: A wire carrying a current of 4 Amperes is having a resistance of 5$\omega\$.Calculate the potential difference across its ends. Basic Electrical Formulas. Electrical designers use it to determine how much voltage is required for a certain load, like a motor, a computer, or even a house full of appliances. Formulas, Tables and Basic Circuits LED CURRENT LED VOLTAGE DROP Forrest M. Mims 111 . If you are on a personal connection, like at home, you can run an anti-virus scan on your device to make sure it is not infected with malware. Volume 1 of 4 Module 1 - Basic Electrical Theory This module describes basic electrical concepts and introduces electrical terminology. Based on the principle of conservation of energy, Kirchhoff’s Current Law (KCL) states that, at any node (junction) in an electrical circuit, the sum of currents flowing into that node is equal to the sum of currents flowing out of that node, and Kirchhoff’s Voltage Law (KVL) states that the sum of all voltage drops around a circuit loop equals zero. Basic Electrical Engineering Pdf Notes – Free BEE Pdf Notes TEXT BOOKS : 1. ELECTRONIC FORMULAS Ohm's Law Formulas for D-C Circuits. If you multiply R (in ohms) by C (in farads), you get what is known as the RC time constant of your RC circuit, symbolized by T: A capacitor charges and discharges almost completely after five times its RC time constant, or 5RC. Solved Examples gives a better understanding of the concepts like current, voltage, resistance etc. The following table presents some common calculations using Ohm’s Law and Joule’s Law. In a resistor-capacitor (RC) circuit, it takes a certain amount of time for the capacitor to charge up to the supply voltage, and then, once fully charged, to discharge down to 0 volts. • Solved Examples gives a better understanding of the concepts like current, voltage, resistance etc. Basic Electrical Fundamentals Voltage, current kilowatts, kilovolt amps. The following table presents some common calculations using Ohm’s Law and Joule’s Law. Electrical and Electronic Technology by hughes Pearson Education. This article deals with electrical formulas and their derivations.