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Connected components of sets and applications 41 and most image processing and analysis operators must therefore be deﬁned on vector images. Deﬁnition 1 Let Ω be a domain (open connected set) in C. We say Ω is simply connected, if the following property holds: no simple closed curve γ in Ω encloses any point of C which is not in Ω. ջ��t��Ƞ�� k1b����죒ws$=���hVj�]6yEV~;�l�k��C�&���O�˵&��4b����N��|��wb�X�8�N�$���©�]v*�����w+�ʴ��[�҉���J35��]mɇ�w���)����'ߑ)�T@о��g�\WI�K���iyECZ YЪ�2(5e���� ��1�xys~�BR�h�I����EҾ�R�@���n�%�FؒL����] z�� �C DڞF3�[�95Jdg�M+ \��e�>tp�Ҩ&��f�X1�4�[�kS�_A-'�%((%�TT XI�g �9(�1���KʢG��oJ���{r*�M��L}~�f�D��=�j�����&���l��-t������U� The union of collection of connected subspaces of X that have a point in common is connected. … If is a continuous function, then is connected. fis connected then what sorts of sets can Date : January 11, 2011. Rachna Choudhary2 Department of Mathematics, University of Delhi, Delhi-110007, India. 22 3. an open, connected set. %PDF-1.5 (O2) If S 1;S 2;:::;S n are open sets, then \n i=1 S i is an open set. Now, the redundancy of the colour images (from the perceptual viewpoint) is high. Astro Pi. De nition 0.1. 7 0 obj << To best describe what is a connected space, we shall describe first what is a disconnected space. The connected subsets of R are exactly intervals or points. 11.2 If f is a path in A from x to y then f[0;1]= f f(t) 2 A j0 t 1 g is called an arc in A that joins x to y. Non-Monge Uniqueness for Independent, Free, Locally Connected Sets S. Brouwer, B. Lindemann, S. Kovalevskaya and Z. In many real world problem we get only partial information about the problem, the vagueness in the description and uncertainty has led to the … Coolest Projects. I.e. Note that while any two separated sets are disjoint, not all disjoint sets are separate. Definition 5.3.1: Connected and Disconnected : An open set S is called disconnected if there are two open, non-empty sets U and V … Locally connected compact sets. Let {A λ}λ∈Λ be a family of connected subsets of a space X. As-sume that each set in this family intersectsS Aλ0 for some λ0 ∈ Λ. Let us assume that Cand D form the … /Length2 9365 It should be noticed that, since / is connected, if A ^0 and B^O, then diX^O. Our whole-day technology showcase celebrating young coders, designers and digital makers around the world. Example 1 (i) Any convex … Connexion . Properties of continuous functions 125 7.3. 1. Retour Produit | Contact : 03 28 41 64 62. A disconnected space is a space that can be separated into two disjoint groups, or more formally: A space ( X , T ) {\displaystyle (X,{\mathcal {T}})} is said to be disconnected iff a pair of disjoint, non-empty open subsets X 1 , X 2 {\displaystyle X_{1},X_{2}} exists, such that X = X 1 ∪ X 2 {\displaystyle X=X_{1}\cup X_{2}} . We will use open sets to divide the set up: DEFN: A set, A ˆX is not connected if there exist open sets U and V such that (i) A ˆU [V (ii) A\U 6= ; (iii) A\V 6= ; (iv) U \V = ; PROBLEM 1: If our metric space X = R with the metric d(x;y) = jx yj. A set E X is said to be connected if E is not the union of two nonempty separated sets. Most recent research has extensively focused on the construction of 1-Connected 1-Dominating Set (1-CDS) in Connected sets 102 5.5. In particular, for any set X, (X;T indiscrete) is connected, as are (R;T ray), (R;T 7) and any other particular point topology on any set, the co-countable and co- nite topologies on uncountable and in nite sets, respectively, etc. Mathematica}, year={2011}, volume={36}, pages={71-80} } L. Rempe; Published 2011; Mathematics ; Annales Academiae Scientiarum Fennicae. Then g :[0;1]! twoway connected yvar xvar as typing. 7. is called connected otherwise. M :t ! I INTRODUCTION . Il y a 1 produit dans votre panier. Thus A is path-connected if and only if, for all x;y 2 A ,x y in A . (Recall that a space is hyperconnected if any pair of nonempty open sets intersect.) ��H��,G?�E�T +%��)��e� ����\���3V2k��ڙ��}N��D�QGV'��9뤙��C�7�{d� When you are ready to proceed, click COMBINE button. 2. is not connected. A connected set is de ned by de ning what it means to be not connected: to be broken into at least two parts. Competitions & Events. Create your user account, with your name and e-mail address. Useful Online Tools. (O3) Let Abe an arbitrary set. Example: [0,1) and [1,2] are disjoint but not separated: [0,1)∩[1,2] = [0,1]∩[1,2] = {1} 6= ∅ [0,1) and … With the new BMW Connected app, the Journey Management features let you set scheduled arrival times for your destinations. Every Design Connected models set is a curated selection of premium 3d models perfectly matched into a ready-to-use arrangement. Expanding and sub-expanding maps. Connected Sets Open Covers and Compactness Suppose (X;d) is a metric space. [Research Report] RR-4597, INRIA. Rechercher. Select up to 20 PDF files and images from your computer or drag them to the drop area. Continuous functions on compact sets 131 7.6. It is the union of all connected sets containing this point. C��^�W�FB���a�|X7枽��%�6�I�!���M�o�L'�lm�NZȃL�{-�v���w�����iEn�2Aa�mn��IWB�P3ܶ����u����}���F����)�\���e[x�� �YG3����c�ִ�z'��0'�V?��,��s��>3L�CN�=- ��Y�(�:���]���}Y(�df��|g5bO�(�Yd�u��̊�H��%��eIg�]*��--,���ssC���������+��0�� ���w,��w��� _6-�"��h�@i E�s��g��E��0�f�ߜ���mc��Z Օ]u.d+�q��a%�Wz___/R�0�R���s����x,!&��{"R葡��aF� Definition A set in in is connected if it is not a subset of the disjoint union of two open sets, both of which it intersects. ����0�������@R$gst��]��υ.\��=b"��r�ġn The next theorem describes the corresponding equivalence relation. A set is a connected total dominating set of the graph G if and only if S admits the following three conditions: a dominating set of graph G, that is for every vertex in is adjacent to at least one vertex in S, equivalently a total dominating set of (Proof: Suppose that X\Y has a point pin it and that Xand Y are connected. Produit ajouté au panier avec succès. if no point of A lies in the closure of B and no point of B lies in the closure of A. 2000 Mathematics Subject Classi cation. We will describe two examples that are subsets of R2. A region D is simply connected if for any z ∈ Dc (the complenent of D) and ε > 0, there is a continuous curve γ(t) with 0 6 t < ∞ such that (i) d(γ(t),Dc) < ε for … Connected Dominating Set (CDS) has been proposed as the virtual backbone to alleviate the broadcasting storm in wireless ad hoc networks. The closure of a connected set is connected. Remark: In other texts, you will see the following equivalent deﬁnition: A set Y in a metric space X is connected if there do not exist open sets A and B such that A∩B = ∅, Y ⊆ A∪B and A∩ Y 6= ∅ and B ∩Y 6= ∅. Proof. 0,00 € Total. #&�Q��DE���΁s�ցu0���c�G�p�i�b��Ԛ�xL�b�:�]��R�Q,�y�X�A�� c�$�T Home Connect Set Up; Setup process in 3 steps; Pairing Overview Video; Appliance installation guides; FAQ... How to connect your home appliance with Home Connect. View Math14958.pdf from MATHS 325 at University of Notre Dame. 11.H. If S is an open set for each 2A, then 2AS is an open set. f((1 t)a + tb … … For the second deﬁnition, observe that any point in the interior of the disc is in Dc and clearly there is no path. Countability Axioms 31 16. @�>D��� D�k����U��M�o?�ё��>{2ј���7�ֿ��ha�7H��\�Ѱ��S�19�Q��81|D��8�>ɭ�z���89�3P����6�2J:��d��3�� G9e*���-.c� J��%�3�����$V1����� t����4p"�9��' %�e�Ҕz[��;4N;$��JrgBC���K4@+p�-�ш��I�0W؅b����ع����A��O01�|Z=*�-�9�a'Ķ����3J�6��D���iq����h��4=/�y�л; 8�4l�J�,� Aregion D is said to be simply connected if any simple closed curve which lies entirely in D can be pulled to a single point in D (a curve is called simple if it has no self intersections). As , ∈ are arbitrary, ∪ is connected. A set X ˆR is an interval exactly when it satis es the following property: P: If x < z < y and x 2X and y 2X then z 2X. The rst one is called the deleted in nite broom. 15 3. <> 11.8 The expressions pathwise-connected and arcwise-connected are often used instead of path-connected . [13] %�쏢 29 0 obj << Connected and Path-connected Spaces 27 14. There is a connected set Kin Xcontaining both aand b. Kis contained in C. Kis the union of disjoint nonempty open sets A\Kand B\K, contradiction.) Open and Closed Sets De nition: A subset Sof a metric space (X;d) is open if it contains an open ball about each of its points | i.e., if 8x2S: 9 >0 : B(x; ) S: (1) Theorem: (O1) ;and Xare open sets. … In topology and related branches of mathematics, a connected space is a topological space that cannot be represented as the union of two or more disjoint non-empty open subsets.Connectedness is one of the principal topological properties that are used to distinguish topological spaces.. A subset of a topological space X is a connected set if it is a connected space when viewed as a subspace of X.. Panier 0 Produit Produits (vide) Aucun produit À définir Livraison 0,00 € Taxes. Quantité . Properties of limits 117 Chapter 7. Proof. Compactness Revisited 30 15. connected. (Careful, this is not the set of all points with both coordinates irrational; it is the set of points such that at least one coordinate is irrational.) A connected set is a set that cannot be partitioned into two nonempty subsets which are open in the relative topology induced on the set.Equivalently, it is a set which cannot be partitioned into two nonempty subsets such that each subset has no points in common with the set closure of the other.. Let be a topological space.A connected set in is a set which cannot be … Non-Monge Uniqueness for Independent, Free, Locally Connected Sets S. Brouwer, B. Lindemann, S. Kovalevskaya and Z. Suppose that there are two nonempty open disjoint sets A and B whose union is X 1 ×X 2. connected sets alpina 118802530/0. Following theorem shows us how to do so. Total . This loop cannot be pulled to a point and hence D is not simply connected. 23 4. 2 M. habib, C. Paul, M. Raﬃnot Trivially, the maximal common connected sets of F form a partition of the vertex set. stream We prove this via contradiction. If , then Since is connected, one of the sets and is empty. For a tree, each pair fu;vgof vertices determines a … Ԏ��b��>�� ���w3�F����k������6���"9��>6��0�)0� �)�=z�ᔚ�v ��Df��W�>^e�Z�Ң��#_���o d�O.Yެ]S��Z�in��. Connexion Mot de passe oublié ? (see hint at … Remark 1 An alternative description of a simply connected do-main is that every closed curve in it can be continuously shrunk to a single point. Upload Files. 11.I. Simple graphs. Components of the interior of K. 17 5. 21 2. �U�{��e�s�A�����lh���-�Tj�X� Computing connected dominated sets with multi-point relays. perfect dominating set, connected total perfect domination number. Continuous functions and open sets 129 7.5. Connected Sets Separated Sets De nition Two subsets A;B of a metric space X are said to be separated if both A \B and A \B are empty. �{�Q٫�W�s��p����L�2b��J��(3v~v�����C�i�6�Z��_g�Nn�9�y�ϛW/^?�|v'�q�BU��\��:�CF� N°1 de la Pièce détachée Motoculture d'Origine. Computing connected dominated sets with multipoint relays Cédric Adjih, Philippe Jacquet, Laurent Viennot To cite this version: Cédric Adjih, Philippe Jacquet, Laurent Viennot. Mathematical Classification: 03E72,05C07, 05C69, 05C72, 05C76. In this rst section, we compare the notion of connectedness in discrete graphs and continuous spaces. Deﬁnition 1.2. "N�I�t���/7��Պ�QOa�����A����~�X��Ə߷fv��@�Wۻ��KЬ3��Sp�����3)�X!Au���?�6���f?�5�^��%)ܩ��H]��_�Y�$����Bf��9Ϫ�U��FF�R�#hVPQ�߳�c�!�t���H��ʲ����#*�}�#4{�4i�F��7���D�N����H��b��i�aubT+��{ȘNc��%�A��^&>�5��$xE��2.����;�ʰ�~w[����ɓ��v���ۛ9��� ��M��4�J����@ ^-�\6"z�.�!h��J�ᙘQ������}��T��+�n�2?c�O�}�Xo.�x=���z� Yd�ɲ����ûA�=HU}. Aucun produit. Chain connected sets in a topological space Zoran Misajleski1 and Aneta Velkoska2 1 Chair of Mathematics, Faculty of Civil Engineering, Ss. endstream CONNECTED SETS OF VAN VLECK PAUL M. SWINGLE Both Vitali and Van Vleck have given interesting constructions of Lebesgue nonmeasurable sets in euclidean Ex. stream Conformal representation of CnKf. x��\I�ܶ�yʧr�c3���/Vr��T�rR�=UNJ�Amq�k���|$����d%)�� o��?߈^������g'�On~�6�_��0)7R��Z�9}t"�or���؍�Was������ngzmE�۳n'���ۋN�:Fc���S���G��Y�s�P6��?�]�^G>��n�m�v��g�s���i'��u~b�Ǉi-��޾�v�W�D��Yf�����Q������D�O�x�^�b�*�z���O'�XT��4X~s� ����rJ?�%+�"�J+��E,dT�#��G�ŗU�]O����alax��R<=�?cӶ¸�/�^��~ܹ^'��qY��,3��b!��t� ���1h"�K��:3eܔ�j�I���&DzNM_;VoMć�n?�"���-��s�Vϴ�� ӱ�~e�d�K� �'������� �j�����U���ɻ���As��-�?F�t.�Pk�T۟:q��f�i��7�M��G�-��z[�h$-&�2\�� �2��>N�{��.��C_ow0A�&��W�b��8��+��8�A&���Rj6�+���}���t8u 3i�b[��EV�D�0n��4������qk�2ҒN;{IRA����B8��QAY(_t�&,���}�>�L�-�ȃ�k��6�Hh0�}�fPe���f�tr"4���v[g�ӽ��L�N}k6�xɸw��Q����u��Al����I���!�5�IUo�R@WHUa�Ԛ ������C�{���ս��+;��B�p�&�̿�.� �;)� Total produits . Drag-and-drop file blocks to change the order. 21 1. xڍ�P�.w-P,�������C A�w�R��)VܝRܵ@�k/�����3�Nf�}^�}^���R�k�JZ��Ar'WVN6!����>'���������V������V�ڂ����B�>�d��φ�'�������'��/�������� �m-�l %��J+ v���Z۸>�������g�� ���Z � �@W���@������_!Dl\]����=<�a�j�AAw�%�7e���56TZ���-�O������ '賋��%x����x�r��X�O�_�p�q��/�߁l��pZX���N^�N� +[൜ ���+ �d��� ?�݁�@�g�?��Ij �������:�B٠��9���fY'Ki��#�����|2���s޽��*������ode�di�����3�����HQ�/�g�2k�+������O r��C��[�������z��=��zC�� �+� ������F��� K[W�9��� ����b�՟���[O�!�s�q8~�Y?w�%����_�?J�.���������RRO�7+/���������^��w�2�7�?��@ۿN���� �?I. Any discrete compact space with more than one element is disconnected. Thus A= X[Y and B= ;.) �4U3I5��N�g�_��M�����ô:���Zo�N߽z?��A�A�pX����~L����n Let pbe a point which is common to all A i.e., p2A . 0 Produit: Produits: 0. 54C60, 54B10, 54D80. %���� A x y f[0;1] 11.3 Suppose that a and b 2 R where a < b and that f :[a ;b ]! M is continuous. Créer un compte. Closed Sets, Hausdor Spaces, and Closure of a Set 9 8. R Sorgenfrey is disconnected. Then X is an interval with … 7. Thus either or is empty. This set is clearly not simply connected. For example, on some systems, the add-on appears when you select Run Without Permission. Basic Point-Set Topology 3 means that f(x) is not in O.On the other hand, x0 was in f −1(O) so f(x 0) is in O.Since O was assumed to be open, there is an interval (c,d) about f(x0) that is contained in O.The points f(x) that are not in O are therefore not in (c,d) so they remain at least a ﬁxed positive distance from f(x0).To summarize: there are points /Length 10382 Projection to a component. Theorem 2.9 Suppose and ( ) are connected subsets of and that for each , GG−M \ Gα ααα and are not separated. Suppose that f : [a;b] !R is a function. Design Connected curated collections are professionally picked combinations of 3d models grouped by style and interesting themes. … Product, Box, and Uniform Topologies 18 11. The most fundamental example of a connected set is the interval [0;1], or more generally any closed or open interval in R. Most reasonable-looking spaces that appear to be connected can be proved to be con-nected using properties of connected sets like the following [2, pp. We need to show that Y = [ A is a connected space. 1 NETWORK COMMUNITIES Network Communities 2 networks are composed of tightly connected sets of [5] M. Demirci, On the convergence structure of L-topological spaces and the Some related … Aide . endobj /Filter /FlateDecode set in a topological space are a union of quasicomponents of the set, and if the set agrees with the space, chain connected components match with quasicomponents. Conversely, if for every open subset U of X, the connected components of U are open, then X admits a base of connected sets and is therefore locally connected. The main property. Connexion. Prove the following: (a) the set A = f1;2gis not connected. 11.7 A set A is path-connected if and only if any two points in A can be joined by an arc in A . Then f is continuous if and only if for every open subset Uof R, f 1(U) is an open subset … If S is an open set for each 2A, then [ 2AS is an open set. (O2) If S 1;S 2;:::;S n are open sets, then \n i=1 S i is an open set. Likewise A\Y = Y. U V = 0; U V = S. A set S (not necessarily open) is called disconnected if there are two open sets U and V such that (U S) # 0 and (V S) # 0(U S) (V S) = 0(U S) (V S) = SIf S is not disconnected it is called connected. On Connected Sublevel Sets in Deep Learning Quynh Nguyen1 Abstract This paper shows that every sublevel set of the loss function of a class of deep over-parameterized neural nets with piecewise linear activation functions is connected and unbounded. Theorem 1.8. A subset K [a;b] is called an open subset of [a;b] if there exists an open set Uof R such that U\[a;b] = K. Proposition 0.2. I��k7d�֦����W ����k�Ƌ������m) K��1Â-_���L �^�Ęr�&1���O��I��5a6f$H �'+�����������+W�L˸��Q�Hk�A���<9�T�����{Oa%�� N�;���E���Գ�c#���K7��ɣ�� NÍrJc/�3����M� 31-���*S�Pyb�E+�����K����[=��>������Ԟ �! Document owners can grant and revoke access to the document, and grant specific readers with specific document access rights. If you do not see the Adobe PDF Reader add-on, try the other options on the menu. This implies that the loss has no bad local valleys and all of its global minima are connected within a unique and potentially very large global valley. Lille 1 CNRS UMR 8022, INRIA Futurs, France {Francois.Ingelrest,David.Simplot}@lifl.fr Ivan Stojmenovic´ Computer Science, SITE University of Ottawa, Canada. /Filter /FlateDecode Carath¶eodory Theorem. 19 7. Cyril and Methodius University-Skopje, misajleski@gf.ukim.edu.mk 2Faculty of Communication Networks and Security, University of Information Science and Technology St.Paul the Apostle - Ohrid, aneta.velkoska@uist.edu.mk In the paper … Continuous Functions 121 7.1. The number of connected sets in G 0containing v is less than the number of connected sets in Gcontaining v. This contradicts the minimality of G, thus proving that Gmust be a tree. Q� H�j�F���B;KWͥ)�����V�&��3�#��UF_��k-��ڧ�%ch�*��v^�}�ؔ:O�8��;�14i2���vᰜkd��߮����wT8�F��� ���kAsP~}W��V�8[뒷���,EQ��.�ė��7W��O�Q�6�/�#/Ah�an4GC�څ��v02�@��Ry+�]���}���P"L����,YH�>erGE�uַB�K�^��#j�7F~���:F�vDK�xjL±)xC�ݛ >[�5��n�#�������W����Kk��F�f�~�'���8�t���II~G���刍�*;�o;U�]69�u,� ����v�����z�'U�m�V�5����-��!m/M Compact Spaces 21 12. In fact any zero dimensional space (that is not indiscrete) is disconnected, as is easy to see. Proof. 11.E. Examples of connected sets that are not path-connected all look weird in some way. Let {Ak}k∈Z be a family of connected sets such that Ak∩Ak+1 6= ∅ for any k ∈ Z. Continuous Functions 12 8.1. Continuity 121 7.2. Open and Closed Sets De nition: A subset Sof a metric space (X;d) is open if it contains an open ball about each of its points | i.e., if 8x2S: 9 >0 : B(x; ) S: (1) Theorem: (O1) ;and Xare open sets. Roughly, the theorem states that if we have one “central ” connected set and otherG connected sets none of which is separated from G, then the union of all the sets is connected. Click the Enable or Disable button (it toggles depending on the status of the selected add-on): Enable sets the Adobe PDF Reader add-on to open PDFs in the browser. %PDF-1.3 16 4. Uniform continuity 127 7.4. Limits 109 6.2. De nition Let E X. Hence there are are m+2 2 connected sets. �\ Ͼ�W᎚�l>]���]��;6S���Ԁ*bw��t�#�ܙF��P�Լ�����rFH�ٳ*[V�E���{�3 The union of two connected sets in a space is connected if the intersection is nonempty. Mathematica; We show that, for many parameters a 2 C, the set … Prove that two points lie in the same … A subset of real line is said to … �Sm��N�z��ʾd�ƠV��KI�Bo{� ��� ����IL�w��o &d�*�z�֗d*H 2�\z��aʌ����d34:������ S*D0�Ǵ��JH�B�YB1�_l�B���%��A�i��R I_���R �3�X+S�'���G� R�w7������~@�}��Z Collections. A set is a connected total dominating set of the graph G if and only if S admits the following three conditions: a dominating set of graph G, that is for every vertex in is adjacent to at least one vertex in S, equivalently a total dominating set of 149{151]: if f: X!Y is continuous and Xis connected then f(X) is connected, if Cis a connected subset of Xthen Cis connected and every set … It is pictured below and consists of the closed line segments L n from (0;0) to (1;1=n) as nruns over the positive integers together with the (red) point (1;0). [12] Similarly x in X , the set P C x {\displaystyle PC_{x}} of all points y such that y ≡ p c x {\displaystyle y\equiv _{pc}x} is called the path component of x . They are called the components or connected components of X. Let A and B be … Das, F uzzy connected sets in fuzzy topological spaces, F uzzy Sets and Systems, 49 (1992), 223-229. As above, with different marker symbols for each set of points twoway connected y1 y2 y3 x, sort msymbol(s d o) ... connected is, in fact, scatter, the difference being that by default the points are connected: Default connect() option: connect(l :::) Thus you get the same results by typing. It is well admitted that the essential geometric features of any natural image are contained in its panchromatic (grey level) representation. Prove that S k∈Z Ak is connected. 11 Path-Connected Sets 11.1 Denition Suppose that A is a non-empty subset of M and that x and y 2 A . Here we give a gen-eralization for the construction of Van Vleck for Em, m^2; our inter-est is in the type of connected set that can be so obtained. 1. Definition 5.3.1: Connected and Disconnected : An open set S is called disconnected if there are two open, non-empty sets U and V such that: . connected. À définir Livraison . stream This free and easy to use online tool allows to combine multiple PDF or images files into a single PDF document without having to install any software. (O3) Let Abe an arbitrary set. C���Y�h6��#��u��~�/���Aee�b_UE1av�n{���F�&�0;1t��)��;������Ь"h8�O 5� �~ ��Z��,D��Z�����ύG�l/"ZqRB ���J���,wv��x�u��_��7 Additionally, the Details /Length3 0 Models Sets. * The Cantor set 104 Chapter 6. ABSTRACT The deﬁnition of -connectedness due to Cs asz r [3] is modiﬁed and its basic properties … Then λ∈Λ Aλ is connected. 18 6. View chap5c_community.pdf from COMP 4211 at The Hong Kong University of Science and Technology. Alternative Definition A set X {\displaystyle X} is called disconnected if there exists a continuous, surjective function f : X → { 0 , 1 } {\displaystyle f:X\to \{0,1\}} , such a function is called a disconnection . Left, right, and in nite limits 114 6.3. Graph theory is one of the most various branches of mathematics with applications to wide variety of subjects. We allow a = 1 ;b = +1. ( ). Key words and phrases. Lemma 1. >> If X[Y is the union of disjoint sets Aand B, both open in A[B, then pbelongs to Aor B, say A. A\Xis open and closed in Xand nonempty, therefore A\X= X. Homeomorphisms 16 10. Il y a 0 produits dans votre panier. Nodal Sets, Multiplicity and Superconductivity in Non-simply Connected Domains Mot de passe. The interconnected objects are represented by points termed as vertices, and the links that connect the vertices are called edges. We have no time to discuss this property deeper. For the ﬁrst deﬁ-nition, take any loop which loops around the unit disc. CONNECTED SETS. On Connected Sublevel Sets in Deep Learning Quynh Nguyen1 Abstract This paper shows that every sublevel set of the loss function of a class of deep over-parameterized neural nets with piecewise linear activation functions is connected and unbounded. In this section we will introduce two other classes of sets: connected and disconnected sets. Allowable trees. A where f(0)= x and f(1)= y is called a path in A from x to y . 20 Chapter 3. A space X {\displaystyle X} that is not disconnected is said to be a connected space. If a set E is a subset of I having property X, then E is not connected. If F only contains a single graph, a maximal common connected set Consider [0;1] and … Two connected components either are disjoint or coincide. First we need to de ne some terms. Then a continuous function f :[0;1]! Step 1: Download the Home Connect app and create your user account. A PROPERTY OF INDECOMPOSABLE CONNECTED SETS 1153 where A =IC\X and B=I — X. 11.9 Throughout this chapter we shall take x y in A to mean there is a path in A from x to y . Elsewhere we will consider the construction of Vitali. Every connected subset of Xis contained in a (unique) component, and the components are the largest connected subsets of X. Connected Sets in R. October 9, 2013 Theorem 1. After finding a destination using the new BMW Connected app, the Details screen will appear displaying the destination, current estimated travel time to arrive there, address, and distance. -Connected Sets B. K. Tyagi Department of Mathematics, Atmaram Sanatan Dharma College, University of Delhi, Delhi-110021, India. Local connectivity of some Julia sets. The abstract concept of a graph is a pair G = (V;E) con-sisting of set of vertices, V, and a set of edges, E, each a pair of vertices. Cuts are sets of vertices or edges whose removal from a graph creates a new graph with more components than 6.Any hyperconnected space is trivially connected. A Theorem of Volterra Vito 15 9. Julia sets. The set of connected vertex sets in a path is in bijection with the set of pairs fu;vgof (not necessarily distinct) vertices. 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Square Channel

Written by Jerry Ratzlaff on . Posted in Plane Geometry

• Three rectangles that intersect at 90° angles at each end of one rectangle.
• A square channel is a structural shape used in construction.

area of a Square Channel formula

 $$\large{ A = w\;l - h \; \left( w - t \right) }$$

Where:

$$\large{ A }$$ = area

$$\large{ h }$$ = height

$$\large{ l }$$ = height

$$\large{ t }$$ = thickness

$$\large{ w }$$ = width

Distance from Centroid of a Square Channel formulas

 $$\large{ C_x = \frac{ 2\;w^2 \;s \;+\; h\;t^2 }{ 2\;w\;l \;-\; 2\;h \; \left( w \;-\; t \right) } }$$ $$\large{ C_y = \frac{ l }{ 2} }$$

Where:

$$\large{ C }$$ = distance from centroid

$$\large{ h }$$ = height

$$\large{ l }$$ = height

$$\large{ s }$$ = thickness

$$\large{ t }$$ = thickness

$$\large{ w }$$ = width

Elastic Section Modulus of a Square Channel formulas

 $$\large{ S_{x} = \frac{ I_{x} }{ C_{y} } }$$ $$\large{ S_{y} = \frac{ I_{y} }{ C_{x} } }$$

Where:

$$\large{ S }$$ = elastic section modulus

$$\large{ C }$$ = distance from centroid

$$\large{ I }$$ = moment of inertia

Perimeter of a Square Channel formula

 $$\large{ P = 2 \; \left( 2\;w + l \; \right) - 2\;t }$$

Where:

$$\large{ P }$$ = perimeter

$$\large{ l }$$ = height

$$\large{ s }$$ = thickness

$$\large{ t }$$ = thickness

$$\large{ w }$$ = width

Polar Moment of Inertia of a Square Channel formulas

 $$\large{ J_{z} = I_{x} + I_{y} }$$ $$\large{ J_{z1} = I_{x1} + I_{y1} }$$

Where:

$$\large{ J }$$ = torsional constant

$$\large{ I }$$ = moment of inertia

Radius of Gyration of a Square Channel formulas

 $$\large{ k_{x} = \sqrt { \frac{ w\;l^3 \;-\; h^3 \; \left( w \;-\; t \right) }{ 12 \; \left[ w\;l \;-\; h \; \left( w \;-\; t \right) \right] } } }$$ $$\large{ k_{y} = \sqrt { \frac{ I_{y} }{ A } } }$$ $$\large{ k_{z} = \sqrt { k_{x}{^2} + k_{y}{^2} } }$$ $$\large{ k_{x1} = \sqrt { \frac{ I_{x1} }{ A } } }$$ $$\large{ k_{y1} = \sqrt { \frac{ I_{y1} }{ A } } }$$ $$\large{ k_{z1} = \sqrt { k_{x1}{^2} + k_{y1}{^2} } }$$

Where:

$$\large{ k }$$ = radius of gyration

$$\large{ A }$$ = area

$$\large{ I }$$ = moment of inertia

$$\large{ h }$$ = height

$$\large{ l }$$ = height

$$\large{ t }$$ = thickness

$$\large{ w }$$ = width

Second Moment of Area of a Square Channel formulas

 $$\large{ I_{x} = \frac{ w\;l^3 \;-\; h^3 \; \left( w \;-\; t \right) }{ 12 } }$$ $$\large{ I_{y} = \frac{ 2\;s\;w^3 \;+\; h\;t^3 }{ 3 } - A\;C_{x}{^2} }$$ $$\large{ I_{x1} = I_{x} + A\;C_{y}{^2} }$$ $$\large{ I_{y1} = I_{y} + A\;C_{x}{^2} }$$

Where:

$$\large{ I }$$ = moment of inertia

$$\large{ A }$$ = area

$$\large{ C }$$ = distance from centroid

$$\large{ h }$$ = height

$$\large{ l }$$ = height

$$\large{ s }$$ = thickness

$$\large{ t }$$ = thickness

$$\large{ w }$$ = width

Torsional Constant of a Square Channel formula

 $$\large{ J = \frac{ 2 \; \left( w \;-\; \frac{t}{2} \right) \; s^3 \; \left( l \;-\; s \right) \; t^3 }{ 3 } }$$

Where:

$$\large{ J }$$ = torsional constant

$$\large{ l }$$ = height

$$\large{ s }$$ = thickness

$$\large{ t }$$ = thickness

$$\large{ w }$$ = width

format: plain


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Surface Integrals Examples 1

Surface Integrals Examples 1

Recall from the Surface Integrals page that the integral of a function $f$ over the surface $\delta$ defined parametrically by $\vec{r}(u, v) = (x(u, v), y(u, v), z(u, v))$ for $(u, v) \in D$ is given by the following formula:

(1)
\begin{align} \quad \quad \iint_{\delta} f(x, y, z) \: dS = \iint_D f(\vec{r}(u, v)) \biggr \| \frac{\partial \vec{r}}{\partial u} \times \frac{\partial \vec{r}}{\partial v} \biggr \| \: du \: dv = \iint_D f(x(u, v), y(u, v), z(u, v)) \sqrt{ \left ( \frac{\partial (y, z)}{\partial (u, v)} \right )^2 + \left ( \frac{\partial (z, x)}{\partial (u, v)} \right )^2 + \left ( \frac{\partial (x, y)}{\partial (u, v)} \right )^2 } \: du \: dv \end{align}

Also recall that if $\delta$ has a one-to-one projection onto the $xy$-plane, and $z = g(x, y)$ (which is equivalent to $G(x, y, z) = z - g(x, y) = 0$) generates our surface $\delta$ then the surface integral of $f$ over $\delta$ can be computed as:

(2)
\begin{align} \quad \iint_{\delta} f(x, y, z) \: dS = \iint_D f(x, y, g(x, y)) \frac{\| \nabla G(x, y, z) \|}{\biggr \| \frac{\partial G}{\partial z} \biggr \|} \: dx \: dy \end{align}

We will now look at some examples of evaluating surface integrals.

Example 1

Evaluate $\iint_{\delta} x \: dS$ where $\delta$ is the parabolic cylinder $z = \frac{x^2}{2}$ that is inside of the circular cylinder $x^2 + y^2 = 1$ in the first octant.

It should not be too difficult to see that the portion of $z = \frac{x^2}{2}$ cut out by the circular cylinder $x^2 + y^2 = 1$ in the first octant has a one-to-one projection onto the $xy$-plane.

Let $G(x, y, z) = z - \frac{x^2}{2} = 0$. Then $\nabla G(x, y, z) = (x, 0, 1)$ and $\frac{\partial G}{\partial z} = 1$. Also, let $D$ represent the quarter disk $x^2 + y^2 ≤ 1$ in the first quadrant of the $xy$-plane. Then we have that:

(3)
\begin{align} \quad \iint_{\delta} x \: dS = \iint_D x \frac{\| \nabla G (x, y, z) \|}{\biggr \| \frac{\partial G}{\partial z} \biggr \|} \: dx \: dy \\ \quad \iint_{\delta} x \: dS = \iint_D x \frac{\sqrt{(x)^2 + (0)^2 + (1)^2}}{\sqrt{(1)^2}} \: dx \: dy \\ \quad \iint_{\delta} x \: dS = \iint_D x \sqrt{x^2 + 1} \: dx \: dy \\ \quad \iint_{\delta} x \: dS = \int_0^1 \int_{0}^{\sqrt{1 - x^2}} x \sqrt{x^2 + 1} \: dy \: dx \\ \quad \iint_{\delta} x \: dS = \int_0^1 x\sqrt{x^2 + 1} \left [ y \right ]_{y = 0}^{y = \sqrt{1 - x^2}} \: dx \\ \quad \iint_{\delta} x \: dS = \int_0^1 x \sqrt{x^2 + 1} \sqrt{1 - x^2} \: dx \\ \quad \iint_{\delta} x \: dS = \int_0^1 x \sqrt{(1 + x^2)(1 - x^2)} \: dx \\ \quad \iint_{\delta} x \: dS = \int_0^1 x \sqrt{1 - x^4} \: dx \end{align}

Now we can evaluate the single integral on the righthand side with substitution. Let $u = x^2$. Then $du = 2x \: dx$ which implies that $\frac{1}{2} \: du = x \: dx$ and so:

(4)
\begin{align} \quad \int x \sqrt{1 - x^4} \: dx = \frac{1}{2} \int \sqrt{1 - u^2} \: du ]]. \end{align} After noting applying the substitution to the limits of integration, notice that we then have the following indefinite integral\quad \iint_{\delta} x \: dS = \frac{1}{2} \int_0^1 \sqrt{1 - u^2} \: du$. This integral is equal to half of the quarter area of the unit circle, that is,$\frac{1}{2} \int_0^1 \sqrt{1 - u^2} \: du = \frac{1}{2} \frac{\pi}{4} = \frac{\pi}{8}$. Example 2 Evaluate$\iint_{\delta} y \: dS$where$\delta$is the portion of the plane$z = 1 + y$that is inside the cone$z = \sqrt{2(x^2 + y^2)}$. Once again, it's not hard to see that the plane$z = 1 + y$has a one-to-one projection onto the$xy$-plane. Let$G(x, y, z) = z - 1 - y$. Then$\nabla G(x, y, z) = (0, -1, 1)$, and$\frac{\partial G}{\partial z} = 1. Thus we have that: (5) \begin{align} \quad dS = \frac{\| \nabla G(x, y, z) \|}{\biggr \| \frac{\partial G}{\partial z} \biggr \|} \: dx \: dy = \frac{\sqrt{(0)^2 + (-1)^2 + (1)^2}}{\sqrt{1^2}} \: dx \: dy = \sqrt{2} \: dx \: dy \end{align} Now note that sincez = 1 + y$and$z = \sqrt{2(x^2 + y^2)}$that in substituting the first equation into the second equation, we get that (by completing the square) the region$D: (6) \begin{align} \quad 1 + y = \sqrt{2(x^2 + y^2)} \\ \quad (1 + y)^2 = 2 (x^2 + y^2) \\ \quad 1 + 2y + y^2 = 2x^2 + 2y^2 \\ \quad 1 = 2x^2 + y^2 - 2y \\ \quad 2 = 2x^2 + y^2 -2y + 1 \\ \quad 2 = 2x^2 + (y - 1)^2 \\ \quad 1 = x^2 + \frac{(y - 1)^2}{2} \end{align} This is an equation of an ellipse centered at(0, 1)$with semi-minor axes having length$1$and with semi-major axes having length$\sqrt{2}$. The area of this ellipse is therefore$\pi (1) (\sqrt{2}) = \pi \sqrt{2}\$, and so:

(7)
\begin{align} \quad \iint_{\delta} y \: dS = \iint_D f(x, y, g(x, y)) \frac{\| \nabla G(x, y, z) \|}{\biggr \| \frac{\partial G}{\partial z} \biggr \|} \: dx \: dy \\ \quad \iint_{\delta} y \: dS = \sqrt{2} \iint_D y \: dx \: dy \\ \quad \iint_{\delta} y \: dS = \sqrt{2} (1) \sqrt{2} \pi \\ \quad \iint_{\delta} y \: dS = 2 \pi \end{align}


text:
# Would a suitably placed inductor stop audio feedback whine?

To give some context, I am designing a cosplay armor costume that somewhat muffles everything that is spoken while wearing it. I have a simple low power audio amp that I was intending to use to amplify my voice and place a small (3") speaker on the outside of the armor, theoretically enabling me to be easily heard outside the outfit without needing to really raise my voice. However, because of the restricted possible range between the mic and the speaker, by the time the volume of audio from the speaker is at acceptable levels, it is evidently picked up again by the mic inside the costume, and produces an annoying squeal from the speaker that rapidly increases in loudness until the amp's volume is turned down, at which point the level of audio from the speaker is undesirably low. The amp is a physically separate device from both the speaker and the (suitably biased) microphone, so I could potentially add some custom electronics to either location. I cannot change any of the properties of the amplifier itself, which takes a line-in, and can directly drive a speaker.

Yes, I know there are commercial stormtrooper amps that likely do most of what I am wanting to accomplish here, but I'd like to see if I can work a solution out myself, and hopefully not have to splurge more than a hundred dollars on what amounts to nothing more than a voice amplifier.

And to that end, I was thinking of possibly using an inductor that coupled with the impedance of either the mic or speaker, depending on inductor placement, would have an L/Z time constant such that the cutoff frequency would be approximately 1.5khz, which is far enough above normal spoken communication that I am hoping it should not interfere with normal voice communication, and still below the frequencies of the squeals produced, which according to the decibel meter that I used to check, peak at well over 2khz.

But I am not sure if that will actually work, nor am I sure whether it would be better to place it on the input or output even if it did.

• Either an inductor in series with the speaker or a RC filter in-line with the mic input.
– user105652
Aug 11 '20 at 3:32
• The main reason is, as you stated, the distance between the speaker and the mic. The mic is able to pick up even the small vibrations that the speaker's diaphragm creates. If there's enough amplification factor then it's inevitable the feedback will occur. So, filtering out above 1.5kHz may not help as the most powerful content is still present. You need to solve this mechanically so that the mic does not pick up the vibrations. I cannot make any further suggestions as I have no idea how the suit and other electronics are placed. Aug 11 '20 at 4:56
• Change to a different electret mic with better far field pickup and only near field within 1” and then place near corner of lips for best pick without breathing noises. Aug 11 '20 at 21:59

I was thinking of possibly using an inductor that coupled with the impedance of either the mic or speaker, depending on inductor placement, would have an L/Z time constant such that the cutoff frequency would be approximately 1.5khz

There is no simple one-size-fits-all solution to this because there can be many, many wavelengths of signal (in the air) between the speaker output and the path audio takes to get back to the microphone. Just moving the speaker or microphone 1 cm can be enough to stop acoustic feedback then, a moment later, due to the audio being reflected back to the microphone along a different path, the signal coming back will sustain oscillation i.e. it becomes positive feedback.

In other words, a simple filter will not do the job effectively at all. It's not like an electronic circuit where negative feedback is applied; it's much more complex than that.

My first recommendation is to incorporate a voice operated switch so that when you are speaking, the amplifier is activated and when you stop speaking closes down thus preventing sustained feedback. This of course relies on normal speech levels being the dominant amplitude.

The next trick is a pitch shifter. I remember electronics magazines way back when publishing a circuit that shifted the whole vocal range by 5 Hz. This is a good way of ensuring that you can't get a sustained feedback but may generate other artefacts. I heard at the time that you could get another 10 dB gain from the amplifier.

I do recall that some megaphones used pitch shift to avoid feedback and I believe that some PA systems do the same.

Along the same lines as pitch-shifting is using a "Dalek effect" "translator" where you are removing a significant amount of natural spectral content of the voice and replacing it with something else. That "something else" can be an assortment of band-limited noise, regular oscillator tones or the output of a musical instrument (aka a vocoder). This type of circuit is generally referred to as a ring-modulator (that's what is used for Dalek voices). I've seen videos of the guy who plays Davros practising lines sat at a table with other Dr, Who actors and wearing the device with no apparent feedback problems.

There are also DSP chips that can eradicate sustained oscillations by modifying in real time the audio filtering algorithm to reduce gain at where it detects the sustained tone.

There is this paper that desribes how frequency-modulation is used to reduce feedback in hearing aids. It considers: -

• Gain reduction techniques
• Frequency shifting
• Adaptive frequency compensation
• Frequency modulation

Elliot Sound Products aka ESP have a very good article on the link called

Acoustic Feedback In PA Systems (With Special Reference To Frequency Shifting)

The article also goes on to show a circuit diagram of a frequency shifter: -

This circuit and another are on the project 204 page.

• I hadn't thought of shifting pitch before. That's actually a good idea, as it might produce a desirable audio effect. I'll look into seeing if I can achieve that with an arduino microntroller, of which I happen to already have several spares that are doing nothing right now.
– Mark
Aug 11 '20 at 17:12


text:
My Math Forum Projection Transformation on x Axis Parallel to y=2x

 Linear Algebra Linear Algebra Math Forum

 October 14th, 2015, 09:57 PM #1 Newbie   Joined: Oct 2015 From: England Posts: 2 Thanks: 0 Projection Transformation on x Axis Parallel to y=2x Hello, Let $T \colon \mathbb{R}^2 \rightarrow \mathbb{R}^2$ be the projection transformation on x axis in parallel to $y=2x$. Find the representing matrix of $T^*$ for the standard basis $B=\left(\left \{1,0 \right \},\left\{0,1\right\}\right)$. I have a solution which I don't understand. A basis for $y=2x$ could be $\left \{(1,2) \right\}$. A basis for x axis could be $\left \{(1,0)\right\}$. Therefore A basis for the whole vector would be $$C=\left\{(1,0)\right\},\left\{(1,2)\right\}$$. I understand why this basis is correct. However: $e_1=(1,0)=1v_1+0v_2 \Rightarrow [e_1]_B=\left (\begin {matrix} 1\\ 0 \end {matrix}\right)$ - this is quite clear. $e_2=(0,1)=-\frac{1}{2}v_1+\frac{1}{2}v_2\Rightarrow [e_2]_B=\left (\begin {matrix} -\frac{1}{2} \\ \frac{1}{2} \end {matrix}\right)$ - this is the part I don't understand. Why use $-\frac{1}{2}$ to multiply $v_1$? What am I missing here? Next, since the proejction transformation is $T(x,y)=(x,0)$ it is easy enough to continue. But there's that part I don't understand. Why is the scalar before the vector $v_1$has to be$-\frac{1} {2}$ Thanks, Alan
 October 15th, 2015, 09:20 AM #2 Banned Camp   Joined: Mar 2015 From: New Jersey Posts: 1,720 Thanks: 126 Given unit vector u. Projection of any vector x onto u in matrix form is given by Tx where $\displaystyle T_{ij} =u_{i}u_{j}$ Change of basis: e1'=P11e1+P21e2 e2'=P12e1+P22e2 Relative to new basis: $\displaystyle T'=P^{-1}TP$ Reference: https://www.khanacademy.org/math/lin...ix-vector-prod Linear Transformations. EDIT: I think I see now what you are asking: Find the projection of x onto x axis in direction of line y=2x. Have to think about how to put that in matrix form. Last edited by zylo; October 15th, 2015 at 09:28 AM.
 October 15th, 2015, 10:08 AM #3 Banned Camp   Joined: Mar 2015 From: New Jersey Posts: 1,720 Thanks: 126 Projection of x in direction u on x axis. ae1+bu=x and solve for a a=x1-x2u1/u2 In matrix form: Tx= $\displaystyle \begin{bmatrix} 1 &-u1/u2 \\ 0& 0 \end{bmatrix}\begin{bmatrix} x1\\ x2 \end{bmatrix}=\begin{bmatrix} x1-x2u1/u2\\ 0 \end{bmatrix}$ Last edited by zylo; October 15th, 2015 at 10:20 AM.
 October 19th, 2015, 09:47 AM #4 Newbie   Joined: Oct 2015 From: England Posts: 2 Thanks: 0 Thank you both!
October 20th, 2015, 08:33 AM   #5
Banned Camp

Joined: Mar 2015
From: New Jersey

Posts: 1,720
Thanks: 126

Quote:
 Originally Posted by Nerazzurri10 Thank you!
You're welcome.

 October 31st, 2015, 07:17 AM #6 Banned Camp   Joined: Mar 2015 From: New Jersey Posts: 1,720 Thanks: 126 Systematic Solution to OP based on first principles. T:Rn->Rm is a linear transformation if T(au+bv)=aTu+bTv Matrix equivalent of T:R2->R2 T(x1e1+x2e2)=x1Te1+x2Te2=x1(T11e1+T21e2)+x2(T12e1+ T22e2) Tij are the matrix components of T Tx=[T]x Example: Project x onto line w in direction v u: unit vector in direction of x v: unit vector in direction of projection w: unit vector along line which x is to be projected onto, au+bv=cw (draw a sketch, all vectors from origin) Te1=cw when au=e1 e1+bv=cw 1+bv1=cw1 bv2=cw2, and solve: c=v2/(w1v2-w2v1) Te2=c'w when a'u=e2 e2+b'v=c'w b'v1=c'e1 1+b'v2=c'w2, and solve c'=-v1/(w1v2-w2v1) Te1=cw=c(w1e1+w2e2)=T11e1+T21e2 Te2=c'w=c'(w1e1+w2e2)=T12e1+T22e2 T11=cw1, T21=cw2, T12=c'w1, T22=c'w2 EDIT: That's the general case. For OP line to be projected onto is x axis so w1=1, w2=0, and also v1=1/sqrt(5), v2=2/sqrt(5) Last edited by zylo; October 31st, 2015 at 07:30 AM.

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text:
## WeBWorK Problems

### Rounding in Random Numbers

by Genevieve Toutain -
Number of replies: 3
I think I posted this in the wrong place yesterday, so I thought I would try it over here.   I am authoring a question that includes a random one digit decimal between -10 and 10, hence:

$a = random(-10,10,.1); I just happened to have seed 1612, and when printing out$a in the problem, instead of printing -0.02 (as I suspect it wanted to), it is printing out -0.1999999999.

I'm having the same issue with

random(-5,5,.1); with seed 4252

printing out -0.899999999999.

Any ideas on what I should do here?  Is there a way to restrict how many decimal places WeBWork displays?  Is this a known issue?  I tried searching the forums to not much avail.

I've gotten around this by replacing it with

Danny


text:
Limit of a Sequence

# Limit of a Sequence

Consider the sequence $\left \{ \frac{1}{n} \right \}_{n=1}^{\infty} = \left \{ 1, \frac{1}{2}, \frac{1}{3}, \frac{1}{4}, ... \right \}$. As $n \to \infty$, it appears as though $\frac{1}{n} \to 0$. In fact, we know that this is true since $\lim_{n \to \infty} \frac{1}{n} = 0$. We will now formalize the definition of a limit with regards to sequences.

 Definition: If $\{ a_n \}$ is a sequence, then $\lim_{n \to \infty} a_n = L$ means that for every $\epsilon > 0$ there exists a corresponding $N \in \mathbb{N}$ such that if $n ≥ N$, then $\mid a_n - L \mid < \epsilon$. If this limit exists, then we say that the sequence $\{ a_n \}$ Converges, and if this limit doesn't exist then we say say the sequence $\{ a_n \}$ Diverges.

We note that our definition of the limit of a sequence is very similar to the limit of a function, in fact, we can think of a sequence as a function whose domain is the set of natural numbers $\mathbb{N}$. From this notion, we obtain the very important theorem:

 Theorem 1: If $\{ a_n \}$ is a sequence and a function $f(x)$, then if $f(n) = a_n$ where $n \in \mathbb{N}$ and $\lim_{x \to \infty} f(x) = L$, then $\lim_{n \to \infty} f(n) = L$.
• Proof of Theorem 1: Let $\epsilon > 0$ be given. We know that $\lim_{x \to \infty} f(x) = L$ which implies that $\forall \epsilon > 0 \: \exists N \in \mathbb{R}$ such that if $x ≥ k$ then $\mid f(x) - L \mid < \epsilon$.
• Now we want to show that $\forall \epsilon > 0 \: \exists N \in \mathbb{N}$ such that if $n ≥ N$ then $\mid a_n - L \mid < \epsilon$. We will choose $N = \mathrm{max} \{ 1, \lceil k \rceil \}$. This ensures that $N$ is an integer.
• Now since $N ≥ k$ then it follows that if $\forall n ≥ N$ then $\mid f(n) - L \mid < \epsilon$. But $f(n) = a_n$ and so $\mid a_n - L \mid < \epsilon$ so $\lim_{n \to \infty} a_n = L$. $\blacksquare$
 Important Note: The converse of this theorem is not implied to be true! That is if $f(n) = a_n$ and $\lim_{n \to \infty} f(n) = L$, then this does NOT imply that $\lim_{x \to \infty} f(x) = L$. For example, consider the sequence $\{ \sin (2\pi n) \} = \{ 0, 0, 0, ... \}$. Clearly this sequence converges at 0. However, the function $f(x) = \sin (2\pi x)$ does not converge, instead, it diverges as it oscillates between $-1$ and $1$.

For example, consider the sequence $\left \{ \frac{n + 2}{n} \right \}_{n=1}^{\infty}$. If we let $f(n) = \frac{n + 2}{n}$ be a function whose domain is the natural numbers, then we calculate the limit of this function like we have in the past, namely:

(1)
\begin{align} \: \lim_{n \to \infty} f(n) = \lim_{n \to \infty} \frac{n + 2}{n} = \lim_{n \to \infty} 1 + \frac{2}{n} = 1 \end{align}

Therefore the limit of our sequence $f(n) = a_n$ is 1, that is, $\left \{ \frac{n + 2}{n} \right \}_{n=1}^{\infty}$ converges to 1 as $n \to \infty$.

Now let's look at another major theorem.

 Theorem 2: If $\lim_{n \to \infty} a_n = L$ and a function $f$ is continuous at $L$, then $\lim_{n \to \infty} f(a_n) = f \left(\lim_{n \to \infty} a_n \right) = f(L)$.
• Proof of Theorem 2: If $f$ is a continuous function at $x = L$, then we know that $f(L)$ is defined and that $\lim_{x \to L} f(x) = f(L)$. By the definition of a limit, $\forall \epsilon > 0 \: \exists \delta > 0$ such that if $0 < \mid x - L \mid < \delta$ then $\mid f(x) - f(L) \mid < \epsilon$. Let $x = a_n$ so then if $0 < \mid a_n - L \mid < \delta$ then $\mid f(a_n) - f(L) \mid < \epsilon$.
• We want to show that $\lim_{n \to \infty} f(a_n) = f(L)$, that is $\forall \epsilon > 0 \: \exists N \in \mathbb{N}$ such that if $n ≥ N$ then $\mid f(a_n) - f(L) \mid < \epsilon$, which is what we showed above.

We will now look at some important limit laws regarding sequences

# Limit Laws of Convergent Sequences

We will now look at some very important limit laws regarding convergent sequences, all of which are analogous to the limit laws for functions that we already know of.

 Law 1 (Addition Law of Convergent Sequences): If the limits of the sequences $\{ a_n \}$ and $\{ b_n \}$ are convergent, that is $\lim_{n \to \infty} a_n = A$ and $\lim_{n \to \infty} b_n = B$ then $\lim_{n \to \infty} (a_n + b_n) = \lim_{n \to \infty} a_n + \lim_{n \to \infty} b_n = A + B$. ( Proof of Law 1 )
 Law 2 (Difference Law of Convergent Sequences): If the limits of the sequences $\{ a_n \}$ and $\{ b_n \}$ are convergent, that is $\lim_{n \to \infty} a_n = A$ and $\lim_{n \to \infty} b_n = B$ then $\lim_{n \to \infty} (a_n - b_n) = \lim_{n \to \infty} a_n - \lim_{n \to \infty} b_n = A - B$. ( Proof of Law 2 )
 Law 3 (Product Law of Convergent Sequences): If the limits of the sequences $\{ a_n \}$ and $\{ b_n \}$ are convergent, that is $\lim_{n \to \infty} a_n = A$ and $\lim_{n \to \infty} b_n = B$, then $\lim_{n \to \infty} (a_nb_n) = \lim_{n \to \infty} a_n \cdot \lim_{n \to \infty} b_n = AB$. ( Proof of Law 3 )
 Law 4 (Quotient Law of Convergent Sequences): If the limits of the sequences $\{ a_n \}$ and $\{ b_n \}$ are convergent, that is $\lim_{n \to \infty} a_n = A$ and $\lim_{n \to \infty} b_n = B$, then $\lim_{n \to \infty} \frac{a_n}{b_n} = \frac{\lim_{n \to \infty} a_n}{\lim_{n \to \infty} b_n} = \frac{A}{B}$ provided that $\lim_{n \to \infty} b_n ≠ 0$. ( Proof of Law 4 )
 Law 5 (Constant Multiple Law of Convergent Sequences): If the limit of the sequence $\{ a_n \}$ is convergent, that is $\lim_{n \to \infty} a_n = A$, and $k$ is a constant, then $\lim_{n \to \infty} ka_n = k \lim_{n \to \infty} a_n = kA$. ( Proof of Law 5 )
 Law 6 (Power Law of Convergent Sequences): If the limit of the sequence $\{ a_n \}$ is convergent, that is $\lim_{n \to \infty} a_n = A$ and $k$ is a non-negative integer, then $\lim_{n \to \infty} (a_n)^k = \left ( \lim_{n \to \infty} a_n \right )^k = (A)^k$ provided that $k \in \mathbb{N}$. ( Proof of Law 6 )
 Law 7 (Squeeze Theorem for Convergent Sequences): If the limits of the sequences $\{ a_n \}$, $\{ b_n \}$, and $\{ c_n \}$ are convergent and $a_n ≤ b_n ≤ c_n$ is true always after some $n^{\mathrm{th}}$ term, if $\lim_{n \to \infty} a_n = \lim_{n \to \infty} c_n = L$, then $\lim_{n \to \infty} b_n = L$. ( Proof of Law 7 )

## Example 1

Determine whether the sequence $\left \{ \frac{x^2 - 1}{x + 1} \right \}_{n=1}^{\infty}$ is convergent or divergent.

Let $f(n) = \frac{n^2 - 1}{n + 1}$ be a function analogous to our sequence. When we factor the numerator and cancel like-terms, we get that $\lim_{n \to \infty} f(n) = \lim_{n \to \infty} \frac{n^2 - 1}{n + 1} = \lim_{n \to \infty} \frac{(n +1)(n - 1)}{n + 1} = \lim_{n \to \infty} n - 1 = \infty$. Therefore the sequence $\left \{ \frac{x^2 - 1}{x + 1} \right \}_{n=1}^{\infty}$ is divergent.


text:
1. ## Exponential model help

I need to solve this problem
2.5649(1.1096^x)=25
And I am not sure how to?
Any help would be greatly appreciated.

2. Hello, JBondman871!

Solve for $\displaystyle x\!:\;\;2.5649(1.1096^x)\:=\:25$

We have: .$\displaystyle 1.1096^x \;=\;\frac{25}{2.5649}$

Take logs: .$\displaystyle \ln\left(1.1096^x\right) \;=\;\ln\left(\frac{25}{2.5649}\right)$

Then: .$\displaystyle x\cdot\ln(1.1096) \;=\;\ln(25) - \ln(2.5649)$

Finally: .$\displaystyle x \;=\;\frac{\ln(25) - \ln(2.5649)}{\ln(1.1096)}$


text:
# FHSST Physics/Electricity/Simple Parallel Circuits

Electricity The Free High School Science Texts: A Textbook for High School Students Studying Physics Main Page - << Previous Chapter (Electrostatics) - Next Chapter (Magnets and Electromagnetism) >> Flow of Charge - Circuits - Voltage and Current - Resistance - Voltage and Current in a Practical Circuit - How Voltage, Current, and Resistance Relate

# Simple parallel circuits

Let's start with a parallel circuit consisting of three resistors and a single battery:

Fhsst_electricity78.png

The first principle to understand about parallel circuits is that the voltage is equal across all components in the circuit. This is because there are only two sets of electrically common points in a parallel circuit, and voltage measured between sets of common points must always be the same at any given time. Therefore, in the above circuit, the voltage across $R_1$ is equal to the voltage across $R_2$ which is equal to the voltage across $R_3$ which is equal to the voltage across the battery. This equality of voltages can be represented in another table for our starting values:

Just as in the case of series circuits, the same caveat for Ohm's Law applies: values for voltage, current, and resistance must be in the same context in order for the calculations to work correctly. However, in the above example circuit, we can immediately apply Ohm's Law to each resistor to find its current because we know the voltage across each resistor (9 volts) and the resistance of each resistor:

Fhsst_electricity80.png

Fhsst_electricity81.png

At this point we still don't know what the total current or total resistance for this parallel circuit is, so we can't apply Ohm's Law to the rightmost ("Total") column. However, if we think carefully about what is happening it should become apparent that the total current must equal the sum of all individual resistor ("branch") currents:

Fhsst_electricity82.png

As the total current exits the negative (-) battery terminal at point 8 and travels through the circuit, some of the flow splits off at point 7 to go up through $R_1$, some more splits off at point 6 to go up through $R_2$, and the remainder goes up through $R_3$. Like a river branching into several smaller streams, the combined flow rates of all streams must equal the flow rate of the whole river. The same thing is encountered where the currents through $R_1$, $R_2$, and $R_3$ join to flow back to the positive terminal of the battery (+) toward point 1: the flow of electrons from point 2 to point 1 must equal the sum of the (branch) currents through $R_1$, $R_2$, and $R_3$.

This is the second principle of parallel circuits: the total circuit current is equal to the sum of the individual branch currents. Using this principle, we can fill in the $I_T$ spot on our table with the sum of $I_{R1}$, $I_{R2}$, and $I_{R3}$:

Finally, applying Ohm's Law to the rightmost ("Total") column, we can calculate the total circuit resistance:

Fhsst_electricity84.png

Please note something very important here. The total circuit resistance is only 625 $\Omega$: less than any one of the individual resistors. In the series circuit, where the total resistance was the sum of the individual resistances, the total was bound to be greater than any one of the resistors individually. Here in the parallel circuit, however, the opposite is true: we say that the individual resistances diminish rather than add to make the total. This principle completes our triad of "rules" for parallel circuits, just as series circuits were found to have three rules for voltage, current, and resistance. Mathematically, the relationship between total resistance and individual resistances in a parallel circuit looks like this:

Fhsst_electricity85.png

The same basic form of equation works for any number of resistors connected together in parallel, just add as many 1/R terms on the denominator of the fraction as needed to accommodate all parallel resistors in the circuit.


text:
# A Special Function Tool-Box for High-Order Finite Element Methods

### Project Description

Subproject of F1301 (FWF-SFB F013)

### Project Duration

01/03/2004 - 29/02/2008

## Publications

### Sum of Squares over Rationals

#### J. Capco, C. Scheiderer

RISC. Technical report, 2019. [url] [pdf]
@techreport{RISC5884,
author = {J. Capco and C. Scheiderer},
title = {{Sum of Squares over Rationals}},
language = {english},
abstract = {Recently it has been shown that a multivariate (homogeneous) polynomialwith rational coefficients that can be written as a sum of squares offorms with real coefficients, is not necessarily a sum of squares offorms with rational coefficients. Essentially, only one constructionfor such forms is known, namely taking the $K/\Q$-norm of a sufficientlygeneral form with coefficients in a number field $K$. Whether thisconstruction yields a form with the desired properties depends onGalois-theoretic properties of $K$ that are not yet well understood.We construct new families of examples, and we shed new light on somewell-known open questions.},
year = {2019},
institution = {RISC},
length = {0},
url = {https://www3.risc.jku.at/~jcapco/public_files/ss18/sosq.html}
}

### On the Complexity of Unsatisfiable Primitive Recursively defined $\Sigma_1$-Sentences

#### David M. Cerna

In: T.B.D, , Proceedings of T.B.D, pp. 1-17. 2019. T.B.D. [pdf]
@inproceedings{RISC5841,
author = {David M. Cerna},
title = {{On the Complexity of Unsatisfiable Primitive Recursively defined $\Sigma_1$-Sentences}},
booktitle = {{T.B.D}},
language = {english},
abstract = {We introduce a measure of complexity based on formula occurrence within instance proofs of an inductive statement. Our measure is closely related to {\em Herbrand Sequent length}, but instead of capturing the number of necessary term instantiations, it captures the finite representational difficulty of a recursive sequence of proofs. We restrict ourselves to a class of unsatisfiable primitive recursively defined negation normal form first-order sentences, referred to as {\em abstract sentences}, which capture many problems of interest; for example, variants of the {\em infinitary pigeonhole principle}. This class of sentences has been particularly useful for inductive formal proof analysis and proof transformation. Together our complexity measure and abstract sentences allow use to capture a notion of {\em tractability} for state-of-the-art approaches to inductive theorem proving, in particular {\em loop discovery} and {\em tree grammar} based inductive theorem provers. We provide a complexity analysis of an important abstract sentence, and discuss the analysis of a few related sentences, based on the infinitary pigeonhole principle which we conjecture represent the upper limits of tractability and foundation of intractability with respect to the current approaches.},
pages = {1--17},
isbn_issn = {T.B.D},
year = {2019},
editor = {T.B.D},
refereed = {yes},
length = {17},
conferencename = {T.B.D}
}

### A Generic Framework for Higher-Order Generalizations

#### David M. Cerna, Temur Kutsia

Technical report no. 19-01 in RISC Report Series, Research Institute for Symbolic Computation (RISC), Johannes Kepler University Linz, Schloss Hagenberg, 4232 Hagenberg, Austria. 2019. [pdf]
@techreport{RISC5883,
author = {David M. Cerna and Temur Kutsia},
title = {{A Generic Framework for Higher-Order Generalizations}},
language = {english},
number = {19-01},
year = {2019},
length = {21},
type = {RISC Report Series},
institution = {Research Institute for Symbolic Computation (RISC), Johannes Kepler University Linz},
address = {Schloss Hagenberg, 4232 Hagenberg, Austria}
}

### Evaluation of the VL Logic (342.208-9) 2018W End of Semester Questionnaire

#### David M. Cerna

Feburary 2019. [pdf] [xlsx]
@techreport{RISC5885,
author = {David M. Cerna},
title = {{Evaluation of the VL Logic (342.208-9) 2018W End of Semester Questionnaire}},
language = {english},
abstract = {In this technical report we cover the choice of layout and intentions behind our end of the semester questionnaire as well as our interpretation of student answers, resulting statistical analysis, and inferences. Our questionnaire is to some extent free-form in that we provide instructions concerning the desired content of the answers but leave the precise formulation of the answer to the student. Our goal, through this approach, was to gain an understanding of how the students viewed there own progress and interest in the course without explicitly guiding them. Towards this end, we chose to have the students draw curves supplemented by short descriptions of important features. We end with a discussion of the benefits and downsides of such a questionnaire as well as what the results entail concerning future iterations of the course. },
year = {2019},
month = {Feburary},
length = {17},
type = {RISC Report Series},
institution = {Research Institute for Symbolic Computation (RISC), Johannes Kepler University Linz},
address = {Schloss Hagenberg, 4232 Hagenberg, Austria}
}

### The Castle Game

#### David M. Cerna

2019. [pdf]
@techreport{RISC5886,
author = {David M. Cerna},
title = {{The Castle Game}},
language = {english},
abstract = {A description of a game for teaching certain aspects of first-order logic based on the Drink's Paradox. },
year = {2019},
length = {3},
type = {RISC Report Series},
institution = {Research Institute for Symbolic Computation (RISC), Johannes Kepler University Linz},
address = {Schloss Hagenberg, 4232 Hagenberg, Austria}
}

### First Order: The Game

#### David M. Cerna

2019. [pdf] [jar]
@techreport{RISC5887,
author = {David M. Cerna},
title = {{First Order: The Game}},
language = {english},
abstract = {A Manual for playing the Game in the Jar file. The game is based on implication based proof situations. This is a demo code for a much larger software project being developed},
year = {2019},
length = {11},
type = {RISC Report Series},
institution = {Research Institute for Symbolic Computation (RISC), Johannes Kepler University Linz},
address = {Schloss Hagenberg, 4232 Hagenberg, Austria}
}

### On the positivity of the Gillis–Reznick–Zeilberger rational function

#### V. Pillwein

Advances in Applied Mathematics 104, pp. 75 - 84. 2019. ISSN 0196-8858. [url]
@article{RISC5813,
author = {V. Pillwein},
title = {{On the positivity of the Gillis–Reznick–Zeilberger rational function}},
language = {english},
abstract = {In this note we provide further evidence for a conjecture of Gillis, Reznick, and Zeilberger on the positivity of the diagonal coefficients of a multivariate rational function. Kauers had proven this conjecture for up to 6 variables using computer algebra. We present a variation of his approach that allows us to prove positivity of the coefficients up to 17 variables using symbolic computation.},
journal = {Advances in Applied Mathematics},
volume = {104},
pages = {75 -- 84},
isbn_issn = { ISSN 0196-8858},
year = {2019},
refereed = {yes},
keywords = {Positivity, Cylindrical decomposition, Rational function, Symbolic summation},
length = {10},
url = {http://www.sciencedirect.com/science/article/pii/S0196885818301179}
}

### Projective and affine symmetries and equivalences of rational and polynomial surfaces

#### M. Hauer, B. Jüttler, J. Schicho

J. Comp. Appl. Math. 349, pp. 424-437. 2019. 0377-0427.
@article{RISC5875,
author = {M. Hauer and B. Jüttler and J. Schicho},
title = {{Projective and affine symmetries and equivalences of rational and polynomial surfaces}},
language = {english},
journal = {J. Comp. Appl. Math.},
volume = {349},
pages = {424--437},
isbn_issn = {0377-0427},
year = {2019},
refereed = {yes},
length = {14}
}

### Towards a symbolic summation theory for unspecified sequences

#### P. Paule, C. Schneider

In: Elliptic Integrals, Elliptic Functions and Modular Forms in Quantum Field Theory, , Texts and Monographs in Symbolic Computation , pp. 351-390. 2019. Springer, ISBN 978-3-030-04479-4. arXiv:1809.06578 [cs.SC]. [url]
@incollection{RISC5750,
author = {P. Paule and C. Schneider},
title = {{Towards a symbolic summation theory for unspecified sequences}},
booktitle = {{Elliptic Integrals, Elliptic Functions and Modular Forms in Quantum Field Theory}},
language = {english},
series = {Texts and Monographs in Symbolic Computation},
pages = {351--390},
publisher = {Springer},
isbn_issn = {ISBN 978-3-030-04479-4},
year = {2019},
note = {arXiv:1809.06578 [cs.SC]},
editor = {J. Blümlein and P. Paule and C. Schneider},
refereed = {yes},
length = {40},
url = {https://arxiv.org/abs/1809.06578}
}

### Numerical Implementation of Harmonic Polylogarithms to Weight w = 8

#### J. Ablinger, J. Blümlein, M. Round, C. Schneider

To appear in Comput. Phys. Comm., pp. 1-19. 2019. ISSN 0010-4655. arXiv:1809.07084 [hep-ph]. [url]
@article{RISC5751,
author = {J. Ablinger and J. Blümlein and M. Round and C. Schneider},
title = {{Numerical Implementation of Harmonic Polylogarithms to Weight w = 8}},
language = {english},
journal = {To appear in Comput. Phys. Comm.},
pages = {1--19},
isbn_issn = {ISSN 0010-4655},
year = {2019},
note = {arXiv:1809.07084 [hep-ph]},
refereed = {yes},
length = {19},
url = {https://arxiv.org/pdf/1809.07084.pdf}
}

### Automated Solution of First Order Factorizable Systems of Differential Equations in One Variable

#### J. Ablinger, J. Blümlein, P. Marquard, N. Rana, C. Schneider

Nucl. Phys. B 939, pp. 253-291. 2019. ISSN 0550-3213. arXiv:1810.12261 [hep-ph]. [url]
@article{RISC5795,
author = {J. Ablinger and J. Blümlein and P. Marquard and N. Rana and C. Schneider},
title = {{Automated Solution of First Order Factorizable Systems of Differential Equations in One Variable}},
language = {english},
journal = {Nucl. Phys. B},
volume = {939},
pages = {253--291},
isbn_issn = {ISSN 0550-3213},
year = {2019},
note = {arXiv:1810.12261 [hep-ph]},
refereed = {yes},
length = {39},
url = {https://www.sciencedirect.com/science/article/pii/S055032131830350X?via%3Dihub}
}

### A Family of Congruences for Rogers-Ramanujan Subpartitions

#### Nicolas Allen Smoot

Journal of Number Theory 196, pp. 35-60. March 2019. ISSN 0022-314X. [pdf]
@article{RISC5809,
author = {Nicolas Allen Smoot},
title = {{A Family of Congruences for Rogers--Ramanujan Subpartitions}},
language = {english},
abstract = {In 2015 Choi, Kim, and Lovejoy studied a weighted partition function, A1(m), which counted subpartitions with a structure related to the Rogers–Ramanujan identities. They conjectured the existence of an infinite class of congruences for A1(m), modulo powers of 5. We give an explicit form of this conjecture, and prove it for all powers of 5.},
journal = {Journal of Number Theory},
volume = {196},
pages = {35--60},
isbn_issn = {ISSN 0022-314X},
year = {2019},
month = {March},
refereed = {yes},
keywords = {Integer partitions, Partition congruences, Rogers--Ramanujan identities, Ramanujan--Kolberg identities, Modular functions},
length = {26}
}

### An Improved Method to Compute the Inverse Mellin Transform of Holonomic Sequences

#### J. Ablinger

In: Proceedings of "Loops and Legs in Quantum Field Theory - LL 2018, , PoS(LL2018) , pp. 1-10. 2018. ISSN 1824-8039. [url]
@inproceedings{RISC5789,
author = {J. Ablinger},
title = {{An Improved Method to Compute the Inverse Mellin Transform of Holonomic Sequences}},
booktitle = {{Proceedings of "Loops and Legs in Quantum Field Theory - LL 2018}},
language = {english},
series = {PoS(LL2018)},
pages = {1--10},
isbn_issn = {ISSN 1824-8039},
year = {2018},
editor = {J. Blümlein and P. Marquard},
refereed = {yes},
length = {10},
url = {https://pos.sissa.it/303/063/pdf}
}

### Anti-Unification and Natural Language Processing

#### N. Amiridze, T. Kutsia

In: Fifth Workshop on Natural Language and Computer Science, NLCS’18, , EasyChair preprints 203, pp. 1-12. 2018. [url] [pdf]
@inproceedings{RISC5707,
author = {N. Amiridze and T. Kutsia},
title = {{Anti-Unification and Natural Language Processing}},
booktitle = {{Fifth Workshop on Natural Language and Computer Science, NLCS’18}},
language = {english},
series = {EasyChair preprints},
number = {203},
pages = {1--12},
isbn_issn = { },
year = {2018},
editor = {A. Asudeh and V. de Paiva and L. Moss},
refereed = {yes},
length = {12},
url = {https://doi.org/10.29007/fkrh}
}

### Varieties of apolar subschemes of toric surfaces

#### Gallet Matteo, Ranestad Kristian, Villamizar Nelly

Ark. Mat. 56(1), pp. 73-99. 2018. ISSN 0004-2080. [url]
@article{RISC5796,
author = {Gallet Matteo and Ranestad Kristian and Villamizar Nelly},
title = {{Varieties of apolar subschemes of toric surfaces}},
language = {english},
journal = {Ark. Mat.},
volume = {56},
number = {1},
pages = {73--99},
isbn_issn = { ISSN 0004-2080},
year = {2018},
refereed = {yes},
length = {27},
url = {https://doi.org/10.4310/ARKIV.2018.v56.n1.a6}
}

### Varieties of apolar subschemes of toric surfaces

#### Gallet Matteo, Ranestad Kristian, Villamizar Nelly

Ark. Mat. 56(1), pp. 73-99. 2018. ISSN 0004-2080. [url]
@article{RISC5811,
author = {Gallet Matteo and Ranestad Kristian and Villamizar Nelly},
title = {{Varieties of apolar subschemes of toric surfaces}},
language = {english},
journal = {Ark. Mat.},
volume = {56},
number = {1},
pages = {73--99},
isbn_issn = { ISSN 0004-2080},
year = {2018},
refereed = {yes},
length = {27},
url = {https://doi.org/10.4310/ARKIV.2018.v56.n1.a6}
}

### Term-Graph Anti-Unification

#### Alexander Baumgartner, Temur Kutsia, Jordi Levy, Mateu Villaret

Technical report no. 18-02 in RISC Report Series, Research Institute for Symbolic Computation (RISC), Johannes Kepler University Linz, Schloss Hagenberg, 4232 Hagenberg, Austria. 2018. [pdf]
@techreport{RISC5549,
author = {Alexander Baumgartner and Temur Kutsia and Jordi Levy and Mateu Villaret},
title = {{Term-Graph Anti-Unification}},
language = {english},
number = {18-02},
year = {2018},
length = {19},
type = {RISC Report Series},
institution = {Research Institute for Symbolic Computation (RISC), Johannes Kepler University Linz},
address = {Schloss Hagenberg, 4232 Hagenberg, Austria}
}

### Term-Graph Anti-Unification

#### Alexander Baumgartner, Temur Kutsia, Jordi Levy, Mateu Villaret

In: 3rd International Conference on Formal Structures for Computation and Deduction (FSCD 2018), , Leibniz International Proceedings in Informatics (LIPIcs) 108, pp. 9:1-9:17. 2018. Schloss Dagstuhl-Leibniz-Zentrum fuer Informatik, Dagstuhl, Germany, ISBN 978-3-95977-077-4 ISSN 1868-8969. [url]
@inproceedings{RISC5764,
author = {Alexander Baumgartner and Temur Kutsia and Jordi Levy and Mateu Villaret},
title = {{Term-Graph Anti-Unification}},
booktitle = {{3rd International Conference on Formal Structures for Computation and Deduction (FSCD 2018)}},
language = {english},
series = {Leibniz International Proceedings in Informatics (LIPIcs)},
volume = {108},
pages = {9:1--9:17},
publisher = {Schloss Dagstuhl--Leibniz-Zentrum fuer Informatik},
isbn_issn = {ISBN 978-3-95977-077-4 ISSN 1868-8969},
year = {2018},
editor = {Helene Kirchner},
refereed = {yes},
length = {17},
url = {http://drops.dagstuhl.de/opus/volltexte/2018/9179}
}

### On some polynomials and series of Bloch-Polya Type

#### Berkovich A., Uncu A. K.

PROCEEDINGS OF THE AMERICAN MATHEMATICAL SOCIETY 146(7), pp. 2827-2838. July 2018. 1088-6826. [url]
@article{RISC5557,
author = {Berkovich A. and Uncu A.~K.},
title = {{On some polynomials and series of Bloch-Polya Type}},
language = {english},
journal = {PROCEEDINGS OF THE AMERICAN MATHEMATICAL SOCIETY},
volume = {146},
number = {7},
pages = {2827--2838},
isbn_issn = {1088-6826},
year = {2018},
month = {July},
refereed = {yes},
keywords = {Mathematics - Number Theory, Mathematics - Combinatorics, 05A17, 05A19, 11B65, 11P81},
length = {12},
url = {http://www.ams.org/journals/proc/2018-146-07/S0002-9939-2018-13982-9/}
}

### Some Elementary Partition Inequalities and Their Implications

#### Berkovich A., Uncu A. K.

ArXiv e-prints (to appear in Annals of Cobinatorics), pp. -. 2018. Preprint. [url]
@article{RISC5558,
author = {Berkovich A. and Uncu A.~K.},
title = {{Some Elementary Partition Inequalities and Their Implications}},
language = {english},
journal = {ArXiv e-prints (to appear in Annals of Cobinatorics)},
pages = {--},
isbn_issn = {Preprint},
year = {2018},
refereed = {yes},
keywords = {Mathematics - Combinatorics, Mathematics - Number Theory, 05A15, 05A17, 05A19, 05A20, 11B65, 11P81, 11P84, 33D15},
length = {12},
url = {https://arxiv.org/abs/1708.01957}
}


text:
# ReproduceIt: Reddit word count

ReproduceIt is a series of articles that reproduce the results from data analysis articles focusing on having open data and open code.

On this small article I reproduce the results (not visualization this time) from the reddit user /u/fhoffa, he posted a nice word cloud visualization of the most common words on some famous subreddits.

The reddit post can be found in the data is beautiful subreddit: Reddit most common words for /r/politics, /r/movies, /r/trees, /r/science and the original word cloud was this:

For some context he mentioned that he used Google BigQuery and Tableau. The data used was the most recent month available (May 2015) in a recent reddit dump that user /u/Stuck_In_the_Matrix made available in a nice torrent.

To reproduce the results I am using dask which is a nice new project from Continuum Analytics which got a lot of attention in the most recent SciPy. A little disclaimer here: I currently work for Continuum but this post is not sponsored in any way.

Dask enables parallel computing through task scheduling and blocked algorithms. This allows developers to write complex parallel algorithms and execute them in parallel either on a modern multi-core machine or on a distributed cluster.

So basically dask is similar to Spark in the way of applying transformation to data but it also works amazingly well in a single multi-core machine and reuses battle tested libraries like numpy and pandas.

The data I am using is the same data as in the original post wich is 33.46 GB of a JSON new line delimited, each line having one comment with some metadata. All this code ran a my MacBook Pro (Retina, Mid 2014) - 2.8 GHz Intel core i7 - 16 GB 1600 MHz DDR3.

Lets jump to the code, as usual some libraries and their versions for reproducibility:

In [1]:
import re
import json
import time
import nltk
import nltk
from nltk.corpus import stopwords

In [2]:
dask.__version__

Out[2]:
'0.6.0'

Note: At the moment I am writing this you need the master version of dask since it includes a recent feature Matt Rocklin recently did but it should be available in the next version of dask.

Update: Dask 0.6.1 has been realeased so you no longer need to use master just conda install dask.

In [4]:
nltk.__version__

Out[4]:
'3.0.4'

Create a data variable that points to the big 33 GB file and set the chunkbytes to 100 MB, map each item to json.loads since I know that each row is a JSON doc.

In [5]:
data = db.from_filenames("RC_2015-05", chunkbytes=100000).map(json.loads)

Similar to spark you can take one or more items to see items of the dataset.

In [6]:
data.take(1)

Out[6]:
({u'archived': False,
u'author': u'rx109',
u'author_flair_css_class': None,
u'author_flair_text': None,
u'controversiality': 0,
u'created_utc': u'1430438400',
u'distinguished': None,
u'downs': 0,
u'edited': False,
u'gilded': 0,
u'id': u'cqug90g',
u'name': u't1_cqug90g',
u'parent_id': u't3_34di91',
u'removal_reason': None,
u'retrieved_on': 1432703079,
u'score': 4,
u'score_hidden': False,
u'subreddit': u'soccer_jp',
u'subreddit_id': u't5_378oi',
u'ups': 4},)

Now some simple helper functions that we are going to use as filters.

In [7]:
no_stopwords = lambda x: x not in stopwords.words('english')

In [8]:
is_word = lambda x: re.search("^[0-9a-zA-Z]+\$", x) is not None

Now it comes the dask part which it will look really similar to anyone who has used Spark.

In [9]:
subreddit = data.filter(lambda x: x['subreddit'] == 'movies')
bodies = subreddit.pluck('body')
words = bodies.map(nltk.word_tokenize).concat()
words2 = words.map(lambda x: x.lower())
words3 = words2.filter(no_stopwords)
words4 = words3.filter(is_word)
counts = words4.frequencies()

Call compute and let dask work. At this moment if you take a look at your process manager you should see something like this:

In [10]:
start_time = time.time()

values = counts.compute()

elapsed_time = time.time() - start_time

We can see how long the computation took, in this case: around 23 minutes which is about 1.4 GB per minute.

In [11]:
elapsed_time  # seconds

Out[11]:
1416.4745790958405

With the word count done we can just sort the python list and see the most common words.

In [12]:
len(values)

Out[12]:
84672
In [13]:
sort = sorted(values, key=lambda x: x[1], reverse=True)

In [14]:
sort[:100]

Out[14]:
[(u'movie', 75848),
(u'like', 60028),
(u'one', 41682),
(u'would', 37354),
(u'think', 33539),
(u'really', 32494),
(u'film', 30931),
(u'movies', 30196),
(u'people', 30046),
(u'good', 27207),
(u'see', 25992),
(u'time', 22783),
(u'get', 21364),
(u'much', 20097),
(u'even', 19977),
(u'deleted', 19464),
(u'could', 19338),
(u'know', 19275),
(u'first', 18435),
(u'well', 18155),
(u'also', 17935),
(u'great', 16729),
(u'make', 16668),
(u'way', 16494),
(u'gt', 16097),
(u'still', 15747),
(u'pretty', 14808),
(u'going', 14649),
(u'watch', 13648),
(u'http', 13338),
(u'actually', 13307),
(u'love', 12916),
(u'something', 12838),
(u'never', 12762),
(u'films', 12688),
(u'seen', 12642),
(u'though', 12576),
(u'character', 12395),
(u'thought', 12332),
(u'say', 12320),
(u'man', 12247),
(u'scene', 12126),
(u'got', 12004),
(u'thing', 11895),
(u'lot', 11864),
(u'go', 11792),
(u'story', 11616),
(u'better', 11599),
(u'want', 11224),
(u'right', 10846),
(u'2', 10698),
(u'back', 10600),
(u'ca', 10434),
(u'action', 10286),
(u'best', 10261),
(u'max', 10066),
(u'point', 10031),
(u'shit', 9546),
(u'sure', 9543),
(u'yeah', 9334),
(u'look', 9181),
(u'new', 9160),
(u'said', 9131),
(u'probably', 8858),
(u'saw', 8832),
(u'every', 8811),
(u'end', 8776),
(u'ever', 8752),
(u'guy', 8717),
(u'world', 8687),
(u'characters', 8605),
(u'two', 8567),
(u'https', 8521),
(u'years', 8478),
(u'feel', 8453),
(u'things', 8200),
(u'many', 8094),
(u'part', 8011),
(u'maybe', 7917),
(u'looks', 7852),
(u'always', 7792),
(u'mean', 7784),
(u'original', 7692),
(u'last', 7643),
(u'big', 7580),
(u'work', 7490),
(u'anything', 7480),
(u'take', 7478),
(u'makes', 7420),
(u'fucking', 7415),
(u'need', 7286),
(u'someone', 7282),
(u'different', 7254),
(u'little', 7241),
(u'whole', 6882),
(u'old', 6782),
(u'kind', 6776)]

Save a file with the results

In [17]:
with open('r_movies.txt', 'w') as f:
for item in sort:
f.write("%s %i\n" % item)

## Conclusions¶

This was a simple post making some computation on 33 GB of data in a single node using around 20 lines of python. The main objective was to reproduce some of the results other people have made with bigger tools like Google Big Query with simpler but also powerful tools in a single node. Also I used dask for the first time and I was kinda impressed by it.

If you have some medium size data a big workstation and dont want to mess up with clusters you should checkout dask. You can also use dask in a cluster but lets save that for a diffent post.


text:
# What is the best way to enter mathematics into the Stack Overflow question body?

I want to be able to enter equations and other mathematics notation into a question body here on Stack Overflow. I haven't found anything about that directly, but I have seen stuff about using HTML for all or part of the question body, and I know there are facilities for entering math into HTML.

Also, I'm very familiar with using LaTeX to describe equations and other math. I suppose it's too much to hope for to wish that I could enter LaTeX directly into questions and answers here on Stack Overflow.

• Nov 3, 2021 at 0:21
• One might argue that if you need the facilities of LaTeX for your question then your question might be more on-topic in one of the forums like math. I won't make that argument myself, there are many times I wished I could use something like LaTeX in one of my SO answers. Nov 3, 2021 at 1:39
• Also relevant: meta.stackoverflow.com/q/396502/6296561 Nov 3, 2021 at 10:13

To enter mathematics on Stack Overflow, we

Also, as mentioned by @PresidentJamesK.Polk, we can copy-and-paste from math symbols and operators in Unicode and insert images into your post. (Images should be used as a last resort because they cannot easily be edited or searched.)

• There are also various Unicode characters that represent mathematical symbols. And finally, in a pinch, you can put in an image. Images should be a last resort. Nov 3, 2021 at 1:35

I prefer using Unicode when possible (and mixed with <sup> and <sub> you can get pretty far), but sometimes you need something heavier.

1. Go to a MathJax-enabled site (eg Mathoverflow) and write your MathJax in some draft. (Don't post.)
2. Take a screenshot of your equation.
3. Insert picture into page. If you insert the MathJax into your post (maybe as a HTML comment) then you can easily make another screenshot if you need to change something

Note: if you do go the route of using Unicode, you can copy it from your draft's rendered MathJax and paste into your post.

One option is to simply include LaTeX code in your post and then tell readers that the equations can be viewed via a userscript or bookmarklet.

As mentioned on meta.SE, there are currently 42 sites on the network which support the MathJax variety of LaTeX. However, the chatrooms associated with those sites do not support MathJax. In those chatrooms it's common to use the JavaScript bookmarklets created by math.SE mod robjohn to render MathJax. You can read about the bookmarks on math.meta.SE, or just go straight to the installation page.

These bookmarklets can render MathJax on any Web page, although the output might not be perfect on some pages. I just did a brief test here using a short \begin{align}...\end{align} sequence that spans several lines, but unfortunately everything got rendered onto a single line.

Another option is CodeCogs. For simple equations, you don't have to use their equation editor, you can simply construct a URL by hand by appending the LaTeX code to the CodeCogs prefix. (Of course, the resulting string needs to be a legal URL). You could specify GIF or PNG for the image format, but Stack Exchange sites can support SVG, which gives a superior result.

Eg, the URL https://latex.codecogs.com/svg.image?x=\frac{-b\pm\sqrt{b^2-4ac}}{2a} gives this result:

$x=\frac{-b\pm\sqrt{b^2-4ac}}{2a}$

I suppose this is an ok option for single equations, but it's a bit tedious if you need multiple lines.


text:
This examples demonstrates how to modify E-Commerce Recommendation template to further adjust score.

By default, items have a weight of 1.0. Giving an item a weight greater than 1.0 will make them appear more often and can be useful for i.e. promoted products. An item can also be given a weight smaller than 1.0 (but bigger than 0), in which case it will be recommended less often than originally. Weight values smaller than 0.0 are invalid.

You can find the complete modified source code here.

## Modification

### ECommAlgorithm.scala

Add a case class to represent each group items which are given the same weight.

 1 2 3 4 5 // ADDED case class WeightGroup( items: Set[String], weight: Double )

In ECommAlgorithm, add weightedItems function to extract the sequence of WeightGroup.

 1 2 3 4 5 6 7 8 9 10 11 12 13 14 15 16 17 18 19 20 21 22 23 24 25 26 27 28  // ADDED /** Get the latest constraint weightedItems */ def weightedItems: Seq[WeightGroup] = { try { val constr = LEventStore.findByEntity( appName = ap.appName, entityType = "constraint", entityId = "weightedItems", eventNames = Some(Seq("$set")), limit = Some(1), latest = true, timeout = Duration(200, "millis") ) if (constr.hasNext) { constr.next.properties.get[Seq[WeightGroup]]("weights") } else { Nil } } catch { case e: scala.concurrent.TimeoutException => logger.error(s"Timeout when read set weightedItems event." + s" Empty list is used.${e}") Nil case e: Exception => logger.error(s"Error when read set weightedItems event: ${e}") throw e } }  Modify the predictKnownUser(), predictDefault() and predictSimilar(): • add the weights: Map[Int, Double] parameter • adjust score according to item weights  1 2 3 4 5 6 7 8 9 10 11 12 13 14 15 16 17 18 19 20 21  def predictKnownUser( userFeature: Array[Double], productModels: Map[Int, ProductModel], query: Query, whiteList: Option[Set[Int]], blackList: Set[Int], weights: Map[Int, Double] // ADDED ): Array[(Int, Double)] = { ... .map { case (i, pm) => // NOTE: features must be defined, so can call .get val s = dotProduct(userFeature, pm.features.get) // may customize here to further adjust score // ADDED val adjustedScore = s * weights(i) (i, adjustedScore) } ... }  Lastly, modify the predict() method. The sequence of WeightGroup transforms into a Map[Int, Double] that we can easily query to extract the weight given to an item, using its Int index.  1 2 3 4 5 6 7 8 9 10 11 12 13 14 15 16 17 18 19 20 21 22 23 24 25 26 27 28 29 30 31 32 33 34 35 36 37 38 39 40 41 42 43 44 45 46 47 48 49 50 51  def predict(model: ECommModel, query: Query): PredictedResult = { ... // ADDED val weights: Map[Int, Double] = (for { group <- weightedItems item <- group.items index <- model.itemStringIntMap.get(item) } yield (index, group.weight)) .toMap .withDefaultValue(1.0) ... val topScores: Array[(Int, Double)] = if (userFeature.isDefined) { // the user has feature vector predictKnownUser( userFeature = userFeature.get, productModels = productModels, query = query, whiteList = whiteList, blackList = finalBlackList, weights = weights // ADDED ) } else { ... if (recentFeatures.isEmpty) { logger.info(s"No features vector for recent items${recentItems}.") predictDefault( productModels = productModels, query = query, whiteList = whiteList, blackList = finalBlackList, weights = weights // ADDED ) } else { predictSimilar( recentFeatures = recentFeatures, productModels = productModels, query = query, whiteList = whiteList, blackList = finalBlackList, weights = weights // ADDED ) } } ... }

Now, to send an event to Event Server:

 1 2 3 4 5 6 7 8 9 10 11 12 13 14 15 16 17 18 19 20 $curl -i -X POST http://localhost:7070/events.json?accessKey=$ACCESS_KEY \ -H "Content-Type: application/json" \ -d '{ "event" : "\$set", "entityType" : "constraint", "entityId" : "weightedItems", "properties" : { "weights": [ { "items": ["i4", "i14"], "weight": 1.2 }, { "items": ["i11"], "weight": 1.5 } ] }, "eventTime" : "2014-11-02T09:39:45.618-08:00" }'


text:
# zbMATH — the first resource for mathematics

On the undecidability of partial polymorphic type reconstruction. (English) Zbl 0789.03017
Summary: We prove that partial type reconstruction for the pure polymorphic $$\lambda$$-calculus is undecidable by a reduction from the second-order unification problem, extending a previous result by H.-J. Boehm. We show further that partial type reconstruction remains undecidable even in a very small predicative fragment of the polymorphic $$\lambda$$-calculus, which implies undecidability of partial type reconstruction for $$\lambda^{ML}$$ as introduced by Harper, Mitchell, and Moggi.

##### MSC:
 03B40 Combinatory logic and lambda calculus 03D35 Undecidability and degrees of sets of sentences


text:
# Problem of the Week #75 - September 2nd, 2013

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#### Chris L T521

##### Well-known member
Staff member
Thanks again to those who participated in last week's POTW! Here's this week's problem!

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Problem
: If $\mathbf{a}$, $\mathbf{b}$ and $\mathbf{c}$ are constant vectors, $\mathbf{r}$ is the position vector $\langle x,y,z\rangle$ and $E$ is given by the inequalities $0\leq \mathbf{a}\cdot\mathbf{r} \leq \alpha$, $0\leq \mathbf{b}\cdot\mathbf{r} \leq \beta$, $0\leq \mathbf{c}\cdot\mathbf{r} \leq \gamma$, show that
$\iiint\limits_E (\mathbf{a}\cdot\mathbf{r}) (\mathbf{b}\cdot\mathbf{r}) (\mathbf{c}\cdot\mathbf{r}) \,dV = \frac{(\alpha \beta \gamma)^2}{8|\mathbf{a}\cdot(\mathbf{b} \times\mathbf{c})|}$

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